LMFDB - Embedded newform 63.6.e.b.37.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,6,Mod(37,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.e (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.1041806482$
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			37.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	63.37
Dual form			63.6.e.b.46.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(16.0000 + 27.7128i) q^{4} +(-105.500 + 75.3442i) q^{7} -427.000 q^{13} +(-512.000 + 886.810i) q^{16} +(-1571.50 + 2721.92i) q^{19} +(1562.50 + 2706.33i) q^{25} +(-3776.00 - 1718.19i) q^{28} +(-1361.50 - 2358.19i) q^{31} +(3330.50 - 5768.60i) q^{37} +22475.0 q^{43} +(5453.50 - 15897.6i) q^{49} +(-6832.00 - 11833.4i) q^{52} +(19313.0 - 33451.1i) q^{61} -32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +(39063.5 + 67660.0i) q^{73} -100576. q^{76} +(-45428.5 + 78684.5i) q^{79} +(45048.5 - 32172.0i) q^{91} -134386. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67}+ \cdots - 268772 q^{97}+O(q^{100}) $ 2 * q + 32 * q^4 - 211 * q^7 - 854 * q^13 - 1024 * q^16 - 3143 * q^19 + 3125 * q^25 - 7552 * q^28 - 2723 * q^31 + 6661 * q^37 + 44950 * q^43 + 10907 * q^49 - 13664 * q^52 + 38626 * q^61 - 65536 * q^64 + 37939 * q^67 + 78127 * q^73 - 201152 * q^76 - 90857 * q^79 + 90097 * q^91 - 268772 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	16.0000	+	27.7128i	0.500000	+	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−105.500	+	75.3442i	−0.813781	+	0.581172i
$7$	−105.500	+	75.3442i	−0.813781	+	0.581172i
$8$		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
$13$	−427.000			−0.700760			−0.350380	−	0.936608i	$-0.613948\pi$
$13$	−427.000			−0.700760			−0.350380	+	0.936608i	$0.613948\pi$
$14$		0			0
$15$		0			0
$16$	−512.000	+	886.810i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−1571.50	+	2721.92i	−0.998689	+	1.72978i	−0.455018	+	0.890482i	$0.650367\pi$
$19$	−1571.50	+	2721.92i	−0.998689	+	1.72978i	−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
$28$	−3776.00	−	1718.19i	−0.910200	−	0.414169i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−1361.50	−	2358.19i	−0.254456	−	0.440731i	0.710291	−	0.703908i	$-0.248563\pi$
$31$	−1361.50	−	2358.19i	−0.254456	−	0.440731i	−0.964748	+	0.263176i	$0.915230\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3330.50	−	5768.60i	0.399949	−	0.692733i	−0.593770	−	0.804635i	$-0.702361\pi$
$37$	3330.50	−	5768.60i	0.399949	−	0.692733i	0.993719	+	0.111902i	$0.0356943\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	22475.0			1.85365			0.926827	−	0.375489i	$-0.122525\pi$
$43$	22475.0			1.85365			0.926827	+	0.375489i	$0.122525\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	5453.50	−	15897.6i	0.324478	−	0.945893i
$50$		0			0
$51$		0			0
$52$	−6832.00	−	11833.4i	−0.350380	−	0.606876i
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	19313.0	−	33451.1i	0.664546	−	1.15103i	−0.314862	−	0.949137i	$-0.601958\pi$
$61$	19313.0	−	33451.1i	0.664546	−	1.15103i	0.979408	−	0.201890i	$-0.0647084\pi$
$62$		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			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	39063.5	+	67660.0i	0.857954	+	1.48602i	0.873877	+	0.486147i	$0.161598\pi$
$73$	39063.5	+	67660.0i	0.857954	+	1.48602i	−0.0159232	+	0.999873i	$0.505069\pi$
$74$		0			0
$75$		0			0
$76$	−100576.			−1.99738
$77$		0			0
$78$		0			0
$79$	−45428.5	+	78684.5i	−0.818956	+	1.41847i	0.0874958	+	0.996165i	$0.472114\pi$
$79$	−45428.5	+	78684.5i	−0.818956	+	1.41847i	−0.906452	+	0.422309i	$0.861220\pi$
$80$		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	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	45048.5	−	32172.0i	0.570265	−	0.407262i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−134386.			−1.45019			−0.725095	−	0.688649i	$-0.758204\pi$
$97$	−134386.			−1.45019			−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
$103$	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		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	123922.	+	214638.i	0.999034	+	1.73038i	0.537567	+	0.843221i	$0.319344\pi$
$109$	123922.	+	214638.i	0.999034	+	1.73038i	0.461467	+	0.887157i	$0.347323\pi$
$110$		0			0
$111$		0			0
$112$	−12800.0	−	132135.i	−0.0964195	−	0.995341i
$113$		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
$121$	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
$127$	347111.			1.90967			0.954837	−	0.297131i	$-0.0960299\pi$
$127$	347111.			1.90967			0.954837	+	0.297131i	$0.0960299\pi$
$128$		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
$133$	−39287.5	−	405566.i	−0.192586	−	1.98807i
$134$		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
$139$	−454657.			−1.99594			−0.997969	−	0.0637074i	$-0.979708\pi$
$139$	−454657.			−1.99594			−0.997969	+	0.0637074i	$0.979708\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	213152.			0.799899
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	204362.	+	353965.i	0.729387	+	1.26333i	0.957143	+	0.289616i	$0.0935277\pi$
$151$	204362.	+	353965.i	0.729387	+	1.26333i	−0.227756	+	0.973718i	$0.573139\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−54607.0	−	94582.1i	−0.176807	−	0.306239i	0.763978	−	0.645242i	$-0.223243\pi$
$157$	−54607.0	−	94582.1i	−0.176807	−	0.306239i	−0.940785	+	0.339004i	$0.889910\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−339124.	+	587380.i	−0.999746	+	1.73161i	−0.480341	+	0.877082i	$0.659487\pi$
$163$	−339124.	+	587380.i	−0.999746	+	1.73161i	−0.519405	+	0.854528i	$0.673846\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−188964.			−0.508935
$170$		0			0
$171$		0			0
$172$	359600.	+	622845.i	0.926827	+	1.60531i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−368750.	−	167792.i	−0.910200	−	0.414169i
$176$		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
$181$	−853027.			−1.93538			−0.967690	−	0.252142i	$-0.918865\pi$
$181$	−853027.			−1.93538			−0.967690	+	0.252142i	$0.918865\pi$
$182$		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
$193$	−328188.	−	568437.i	−0.634204	−	1.09847i	−0.986683	−	0.162653i	$-0.947995\pi$
$193$	−328188.	−	568437.i	−0.634204	−	1.09847i	0.352480	−	0.935820i	$-0.385339\pi$
$194$		0			0
$195$		0			0
$196$	527824.	−	103230.i	0.981407	−	0.191941i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	456134.	+	790047.i	0.816507	+	1.41423i	0.908241	+	0.418448i	$0.137426\pi$
$199$	456134.	+	790047.i	0.816507	+	1.41423i	−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
$208$	218624.	−	378668.i	0.350380	−	0.606876i
$209$		0			0
$210$		0			0
$211$	−288976.			−0.446844			−0.223422	−	0.974722i	$-0.571723\pi$
$211$	−288976.			−0.446844			−0.223422	+	0.974722i	$0.571723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	321314.	+	146208.i	0.463213	+	0.210776i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	304052.			0.409436			0.204718	−	0.978821i	$-0.434372\pi$
$223$	304052.			0.409436			0.204718	+	0.978821i	$0.434372\pi$
$224$		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
$229$	729754.	−	1.26397e6i	0.919576	−	1.59275i	0.119515	−	0.992832i	$-0.461866\pi$
$229$	729754.	−	1.26397e6i	0.919576	−	1.59275i	0.800060	−	0.599919i	$-0.204801\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−648487.	−	1.12321e6i	−0.719215	−	1.24572i	−0.961311	−	0.275465i	$-0.911168\pi$
$241$	−648487.	−	1.12321e6i	−0.719215	−	1.24572i	0.242096	−	0.970252i	$-0.422165\pi$
$242$		0			0
$243$		0			0
$244$	1.23603e6			1.32909
$245$		0			0
$246$		0			0
$247$	671030.	−	1.16226e6i	0.699842	−	1.21216i
$248$		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
$256$	−524288.	−	908093.i	−0.500000	−	0.866025i
$257$		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	+	859521.i	0.0771259	+	0.796172i
$260$		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
$268$	−607024.	+	1.05140e6i	−0.516260	+	0.894189i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−1.12643e6	+	1.95103e6i	−0.931707	+	1.61376i	−0.151304	+	0.988487i	$0.548347\pi$
$271$	−1.12643e6	+	1.95103e6i	−0.931707	+	1.61376i	−0.780403	+	0.625277i	$0.784986\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−369194.	−	639464.i	−0.289105	−	0.500745i	0.684491	−	0.729021i	$-0.260024\pi$
$277$	−369194.	−	639464.i	−0.289105	−	0.500745i	−0.973596	+	0.228276i	$0.926691\pi$
$278$		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
$283$	1.16729e6	+	2.02180e6i	0.866387	+	1.50063i	0.865663	+	0.500627i	$0.166897\pi$
$283$	1.16729e6	+	2.02180e6i	0.866387	+	1.50063i	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
$292$	−1.25003e6	+	2.16512e6i	−0.857954	+	1.48602i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−2.37111e6	+	1.69336e6i	−1.50847	+	1.07729i
$302$		0			0
$303$		0			0
$304$	−1.60922e6	−	2.78724e6i	−0.998689	−	1.72978i
$305$		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		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−1.28354e6	+	2.22315e6i	−0.740539	+	1.28265i	0.211712	+	0.977332i	$0.432096\pi$
$313$	−1.28354e6	+	2.22315e6i	−0.740539	+	1.28265i	−0.952250	+	0.305318i	$0.901237\pi$
$314$		0			0
$315$		0			0
$316$	−2.90742e6			−1.63791
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−667188.	−	1.15560e6i	−0.350380	−	0.606876i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	273076.	−	472981.i	0.136998	−	0.237287i	−0.789361	−	0.613929i	$-0.789588\pi$
$331$	273076.	−	472981.i	0.136998	−	0.237287i	0.926359	+	0.376642i	$0.122921\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.63172e6			1.26231			0.631155	−	0.775657i	$-0.282581\pi$
$337$	2.63172e6			1.26231			0.631155	+	0.775657i	$0.282581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	622450.	+	2.08809e6i	0.285673	+	0.958327i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	4.27561e6			1.87904			0.939518	−	0.342501i	$-0.111274\pi$
$349$	4.27561e6			1.87904			0.939518	+	0.342501i	$0.111274\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−3.70118e6	−	6.41062e6i	−1.49476	−	2.58900i
$362$		0			0
$363$		0			0
$364$	1.61235e6	+	733669.i	0.637832	+	0.290233i
$365$		0			0
$366$		0			0
$367$	1.29159e6	+	2.23710e6i	0.500563	+	0.867000i	1.00000		0.000650122i	$0.000206940\pi$
$367$	1.29159e6	+	2.23710e6i	0.500563	+	0.867000i	−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
$379$	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
$388$	−2.15018e6	−	3.72421e6i	−0.725095	−	1.25590i
$389$		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
$397$	2.50171e6	−	4.33309e6i	0.796638	−	1.37982i	−0.125156	−	0.992137i	$-0.539943\pi$
$397$	2.50171e6	−	4.33309e6i	0.796638	−	1.37982i	0.921794	−	0.387681i	$-0.126724\pi$
$398$		0			0
$399$		0			0
$400$	−3.20000e6			−1.00000
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	581361.	+	1.00695e6i	0.178313	+	0.308847i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.58510e6	+	4.47753e6i	0.764134	+	1.32352i	0.940703	+	0.339231i	$0.110167\pi$
$409$	2.58510e6	+	4.47753e6i	0.764134	+	1.32352i	−0.176569	+	0.984288i	$0.556500\pi$
$410$		0			0
$411$		0			0
$412$	6.76726e6			1.96413
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		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
$427$	482825.	+	4.98421e6i	0.128150	+	1.32290i
$428$		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
$433$	−7.49192e6			−1.92032			−0.960160	−	0.279450i	$-0.909848\pi$
$433$	−7.49192e6			−1.92032			−0.960160	+	0.279450i	$0.909848\pi$
$434$		0			0
$435$		0			0
$436$	−3.96549e6	+	6.86843e6i	−0.999034	+	1.73038i
$437$		0			0
$438$		0			0
$439$	−1.99667e6	+	3.45833e6i	−0.494475	+	0.856455i	−0.999980	−	0.00636830i	$-0.997973\pi$
$439$	−1.99667e6	+	3.45833e6i	−0.494475	+	0.856455i	0.505505	+	0.862824i	$0.331306\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.45702e6	−	2.46888e6i	0.813781	−	0.581172i
$449$		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
$457$	−592880.	+	1.02690e6i	−0.132793	+	0.230005i	−0.924752	−	0.380569i	$-0.875728\pi$
$457$	−592880.	+	1.02690e6i	−0.132793	+	0.230005i	0.791959	+	0.610574i	$0.209061\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	2.92217e6			0.633510			0.316755	−	0.948507i	$-0.397407\pi$
$463$	2.92217e6			0.633510			0.316755	+	0.948507i	$0.397407\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	−4.47680e6	−	2.03708e6i	−0.939801	−	0.427638i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.82188e6			−1.99738
$476$		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
$484$	5.15363e6			1.00000
$485$		0			0
$486$		0			0
$487$	682437.	+	1.18202e6i	0.130389	+	0.225840i	0.923827	−	0.382811i	$-0.125044\pi$
$487$	682437.	+	1.18202e6i	0.130389	+	0.225840i	−0.793438	+	0.608652i	$0.791711\pi$
$488$		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
$496$	2.78835e6			0.508913
$497$		0			0
$498$		0			0
$499$	1.08685e6	−	1.88247e6i	0.195397	−	0.338437i	−0.751634	−	0.659581i	$-0.770734\pi$
$499$	1.08685e6	−	1.88247e6i	0.195397	−	0.338437i	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
$508$	5.55378e6	+	9.61942e6i	0.954837	+	1.65383i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−9.21899e6	−	4.19492e6i	−1.56182	−	0.710675i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	5.86434e6	−	1.01573e7i	0.937486	−	1.62377i	0.167346	−	0.985898i	$-0.446480\pi$
$523$	5.86434e6	−	1.01573e7i	0.937486	−	1.62377i	0.770140	−	0.637875i	$-0.220186\pi$
$524$		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
$532$	1.06108e7	−	7.57782e6i	1.62543	−	1.16082i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	−0.733426	−	0.679769i	$-0.762080\pi$
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	0.955410	+	0.295281i	$0.0954134\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.27982e7			1.82886			0.914430	−	0.404744i	$-0.132639\pi$
$547$	1.27982e7			1.82886			0.914430	+	0.404744i	$0.132639\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.13571e6	−	1.17240e7i	−0.157927	−	1.63028i
$554$		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
$559$	−9.59682e6			−1.29897
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	−2.94390e6	−	5.09898e6i	−0.377862	−	0.654475i	0.612889	−	0.790169i	$-0.290007\pi$
$571$	−2.94390e6	−	5.09898e6i	−0.377862	−	0.654475i	−0.990751	+	0.135693i	$0.956674\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.45496e6	+	1.11803e7i	0.807150	+	1.39802i	0.914830	+	0.403839i	$0.132324\pi$
$577$	6.45496e6	+	1.11803e7i	0.807150	+	1.39802i	−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			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	8.55839e6			1.01649
$590$		0			0
$591$		0			0
$592$	3.41043e6	+	5.90704e6i	0.399949	+	0.692733i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−1.74342e7			−1.96887			−0.984435	−	0.175749i	$-0.943765\pi$
$601$	−1.74342e7			−1.96887			−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
$607$	7.84027e6	−	1.35797e7i	0.863693	−	1.49596i	−0.00464665	−	0.999989i	$-0.501479\pi$
$607$	7.84027e6	−	1.35797e7i	0.863693	−	1.49596i	0.868339	−	0.495970i	$-0.165188\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7.91172e6	+	1.37035e7i	0.850394	+	1.47292i	0.880853	+	0.473389i	$0.156969\pi$
$613$	7.91172e6	+	1.37035e7i	0.850394	+	1.47292i	−0.0304599	+	0.999536i	$0.509697\pi$
$614$		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
$619$	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
$628$	1.74742e6	−	3.02663e6i	0.176807	−	0.306239i
$629$		0			0
$630$		0			0
$631$	−1.62439e7			−1.62411			−0.812057	−	0.583579i	$-0.801652\pi$
$631$	−1.62439e7			−1.62411			−0.812057	+	0.583579i	$0.801652\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.32864e6	+	6.78829e6i	−0.227381	+	0.662844i
$638$		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
$643$	−1.33848e7			−1.27668			−0.638342	−	0.769753i	$-0.720380\pi$
$643$	−1.33848e7			−1.27668			−0.638342	+	0.769753i	$0.720380\pi$
$644$		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
$652$	−2.17039e7			−1.99949
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		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
$673$	1.73115e7			1.47332			0.736661	−	0.676262i	$-0.236401\pi$
$673$	1.73115e7			1.47332			0.736661	+	0.676262i	$0.236401\pi$
$674$		0			0
$675$		0			0
$676$	−3.02342e6	−	5.23672e6i	−0.254467	−	0.440751i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	1.41777e7	−	1.01252e7i	1.18014	−	0.842810i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.15072e7	+	1.99311e7i	−0.926827	+	1.60531i
$689$		0			0
$690$		0			0
$691$	−3.50960e6	+	6.07881e6i	−0.279616	+	0.484310i	−0.971289	−	0.237901i	$-0.923541\pi$
$691$	−3.50960e6	+	6.07881e6i	−0.279616	+	0.484310i	0.691673	+	0.722211i	$0.256874\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.25000e6	−	1.29038e7i	−0.0964195	−	0.995341i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	1.04678e7	+	1.81307e7i	0.798850	+	1.38365i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.13507e7	−	1.96600e7i	0.848021	−	1.46882i	−0.0349502	−	0.999389i	$-0.511127\pi$
$709$	1.13507e7	−	1.96600e7i	0.848021	−	1.46882i	0.882971	−	0.469427i	$-0.155539\pi$
$710$		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
$721$	2.64346e6	+	2.72885e7i	0.189380	+	1.95498i
$722$		0			0
$723$		0			0
$724$	−1.36484e7	−	2.36398e7i	−0.967690	−	1.67609i
$725$		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
$733$	8.78241e6	−	1.52116e7i	0.603746	−	1.04572i	−0.388503	−	0.921448i	$-0.627008\pi$
$733$	8.78241e6	−	1.52116e7i	0.603746	−	1.04572i	0.992248	−	0.124270i	$-0.0396590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.40852e6	−	1.10999e7i	−0.431665	−	0.747666i	0.565352	−	0.824850i	$-0.308740\pi$
$739$	−6.40852e6	−	1.10999e7i	−0.431665	−	0.747666i	−0.997017	+	0.0771842i	$0.975407\pi$
$740$		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
$751$	1.36610e7	−	2.36615e7i	0.883857	−	1.53088i	0.0368381	−	0.999321i	$-0.488271\pi$
$751$	1.36610e7	−	2.36615e7i	0.883857	−	1.53088i	0.847019	−	0.531563i	$-0.178395\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.98526e7			1.25915			0.629575	−	0.776940i	$-0.283229\pi$
$757$	1.98526e7			1.25915			0.629575	+	0.776940i	$0.283229\pi$
$758$		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
$763$	−2.92455e7	−	1.33076e7i	−1.81864	−	0.827537i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.50017e7			1.52459			0.762296	−	0.647228i	$-0.224072\pi$
$769$	2.50017e7			1.52459			0.762296	+	0.647228i	$0.224072\pi$
$770$		0			0
$771$		0			0
$772$	1.05020e7	−	1.81900e7i	0.634204	−	1.09847i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	4.25469e6	−	7.36933e6i	0.254456	−	0.440731i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.13060e7	+	1.29758e7i	0.656929	+	0.753953i
$785$		0			0
$786$		0			0
$787$	−1.65531e7	−	2.86708e7i	−0.952668	−	1.65007i	−0.739616	−	0.673029i	$-0.764993\pi$
$787$	−1.65531e7	−	2.86708e7i	−0.952668	−	1.65007i	−0.213052	−	0.977041i	$-0.568340\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8.24665e6	+	1.42836e7i	−0.465688	+	0.806595i
$794$		0			0
$795$		0			0
$796$	−1.45963e7	+	2.52815e7i	−0.816507	+	1.41423i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$	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
$817$	−3.53195e7	+	6.11751e7i	−1.85122	+	3.20641i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	−1.11462e7	−	1.93057e7i	−0.573623	−	0.993543i	−0.996190	−	0.0872118i	$-0.972204\pi$
$823$	−1.11462e7	−	1.93057e7i	−0.573623	−	0.993543i	0.422567	−	0.906332i	$-0.361129\pi$
$824$		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
$829$	1.13243e7	+	1.96142e7i	0.572299	+	0.991252i	0.996329	+	0.0856034i	$0.0272818\pi$
$829$	1.13243e7	+	1.96142e7i	0.572299	+	0.991252i	−0.424030	+	0.905648i	$0.639385\pi$
$830$		0			0
$831$		0			0
$832$	1.39919e7			0.700760
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−2.05111e7			−1.00000
$842$		0			0
$843$		0			0
$844$	−4.62362e6	−	8.00834e6i	−0.223422	−	0.386978i
$845$		0			0
$846$		0			0
$847$	2.01314e6	+	2.07817e7i	0.0964195	+	0.995341i
$848$		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		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−1.53387e6	+	2.65674e6i	−0.0709259	+	0.122847i	−0.899307	−	0.437317i	$-0.855929\pi$
$859$	−1.53387e6	+	2.65674e6i	−0.0709259	+	0.122847i	0.828381	+	0.560164i	$0.189262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	1.08920e6	+	1.12438e7i	0.0490691	+	0.506542i
$869$		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
$877$	1.91966e6	−	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			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−4.58066e7			−1.97709			−0.988545	−	0.150925i	$-0.951775\pi$
$883$	−4.58066e7			−1.97709			−0.988545	+	0.150925i	$0.951775\pi$
$884$		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
$889$	−3.66202e7	+	2.61528e7i	−1.55406	+	1.10985i
$890$		0			0
$891$		0			0
$892$	4.86483e6	+	8.42614e6i	0.204718	+	0.354582i
$893$		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			−	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
$916$	4.67042e7			1.83915
$917$		0			0
$918$		0			0
$919$	2.10911e6	−	3.65308e6i	0.0823777	−	0.142682i	−0.821893	−	0.569642i	$-0.807082\pi$
$919$	2.10911e6	−	3.65308e6i	0.0823777	−	0.142682i	0.904271	+	0.426959i	$0.140415\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.08156e7			0.799899
$926$		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
$931$	3.47019e7	+	3.98271e7i	1.31214	+	1.50593i
$932$		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	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	−1.66801e7	−	2.88908e7i	−0.601220	−	1.04134i
$950$		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
$961$	1.06072e7	−	1.83722e7i	0.370504	−	0.641732i
$962$		0			0
$963$		0			0
$964$	2.07516e7	−	3.59428e7i	0.719215	−	1.24572i
$965$		0			0
$966$		0			0
$967$	−3.23453e7			−1.11236			−0.556180	−	0.831062i	$-0.687733\pi$
$967$	−3.23453e7			−1.11236			−0.556180	+	0.831062i	$0.687733\pi$
$968$		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	−	3.42558e7i	1.62426	−	1.15998i
$974$		0			0
$975$		0			0
$976$	1.97765e7	+	3.42539e7i	0.664546	+	1.15103i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	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
$988$	4.29460e7			1.39968
$989$		0			0
$990$		0			0
$991$	−3.02764e7	−	5.24403e7i	−0.979310	−	1.69622i	−0.664908	−	0.746925i	$-0.731529\pi$
$991$	−3.02764e7	−	5.24403e7i	−0.979310	−	1.69622i	−0.314402	−	0.949290i	$-0.601804\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.02364e7	−	5.23710e7i	−0.963368	−	1.66860i	−0.713937	−	0.700210i	$-0.753090\pi$
$997$	−3.02364e7	−	5.23710e7i	−0.963368	−	1.66860i	−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	63.6.e.b.37.1	✓	2
3.2	odd	2	CM	63.6.e.b.37.1	✓	2
7.2	even	3		441.6.a.e.1.1		1
7.4	even	3	inner	63.6.e.b.46.1	yes	2
7.5	odd	6		441.6.a.f.1.1		1
21.2	odd	6		441.6.a.e.1.1		1
21.5	even	6		441.6.a.f.1.1		1
21.11	odd	6	inner	63.6.e.b.46.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.e.b.37.1	✓	2	1.1	even	1	trivial
63.6.e.b.37.1	✓	2	3.2	odd	2	CM
63.6.e.b.46.1	yes	2	7.4	even	3	inner
63.6.e.b.46.1	yes	2	21.11	odd	6	inner
441.6.a.e.1.1		1	7.2	even	3
441.6.a.e.1.1		1	21.2	odd	6
441.6.a.f.1.1		1	7.5	odd	6
441.6.a.f.1.1		1	21.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(16.0000 + 27.7128i) q^{4} +(-105.500 + 75.3442i) q^{7} -427.000 q^{13} +(-512.000 + 886.810i) q^{16} +(-1571.50 + 2721.92i) q^{19} +(1562.50 + 2706.33i) q^{25} +(-3776.00 - 1718.19i) q^{28} +(-1361.50 - 2358.19i) q^{31} +(3330.50 - 5768.60i) q^{37} +22475.0 q^{43} +(5453.50 - 15897.6i) q^{49} +(-6832.00 - 11833.4i) q^{52} +(19313.0 - 33451.1i) q^{61} -32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +(39063.5 + 67660.0i) q^{73} -100576. q^{76} +(-45428.5 + 78684.5i) q^{79} +(45048.5 - 32172.0i) q^{91} -134386. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67}+ \cdots - 268772 q^{97}+O(q^{100}) \) 2 * q + 32 * q^4 - 211 * q^7 - 854 * q^13 - 1024 * q^16 - 3143 * q^19 + 3125 * q^25 - 7552 * q^28 - 2723 * q^31 + 6661 * q^37 + 44950 * q^43 + 10907 * q^49 - 13664 * q^52 + 38626 * q^61 - 65536 * q^64 + 37939 * q^67 + 78127 * q^73 - 201152 * q^76 - 90857 * q^79 + 90097 * q^91 - 268772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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