LMFDB - Embedded newform 177.6.d.a.176.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,6,Mod(176,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("177.176");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, 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:	$28.3879361069$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.149721291.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{3} + 27 $ x^6 - 7*x^3 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			176.3
Root			$1.66591 - 0.474089i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.6.d.a.176.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +O(q^{10})$	$q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +(-91.5268 + 490.362i) q^{12} +(764.854 + 142.761i) q^{15} +1024.00 q^{16} -683.622i q^{17} +896.052 q^{19} -1597.21i q^{20} +(-739.692 + 3962.96i) q^{21} +633.714 q^{25} +(-1991.50 + 3222.24i) q^{27} +8275.66 q^{28} +3246.37i q^{29} -12908.2i q^{35} +(7252.43 + 2805.08i) q^{36} -18900.1i q^{41} +(4375.29 - 11312.2i) q^{45} +(2928.86 - 15691.6i) q^{48} +50074.3 q^{49} +(-10475.7 - 1955.30i) q^{51} +38850.3i q^{53} +(2562.90 - 13730.9i) q^{57} -26738.1i q^{59} +(-24475.3 - 4568.36i) q^{60} +(58611.9 + 22669.8i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(1812.56 - 9710.91i) q^{75} -28673.7 q^{76} +96849.2 q^{79} +51110.7i q^{80} +(43680.9 + 39733.7i) q^{81} +(23670.1 - 126815. i) q^{84} +34121.5 q^{85} +(49746.7 + 9285.31i) q^{87} +44724.5i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 192 q^{4}+O(q^{10}) $ 6 * q - 192 * q^4	$ 6 q - 192 q^{4} + 3153 q^{15} + 6144 q^{16} - 3075 q^{21} - 18750 q^{25} - 11949 q^{27} + 81147 q^{45} + 100842 q^{49} - 131001 q^{57} - 100896 q^{60} + 180732 q^{63} - 196608 q^{64} - 197304 q^{75} + 98400 q^{84} + 221799 q^{87}+O(q^{100}) $ 6 * q - 192 * q^4 + 3153 * q^15 + 6144 * q^16 - 3075 * q^21 - 18750 * q^25 - 11949 * q^27 + 81147 * q^45 + 100842 * q^49 - 131001 * q^57 - 100896 * q^60 + 180732 * q^63 - 196608 * q^64 - 197304 * q^75 + 98400 * q^84 + 221799 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	2.86021	−	15.3238i	0.183483	−	0.983023i
$3$	2.86021	−	15.3238i	0.183483	−	0.983023i
$4$	−32.0000			−1.00000
$5$			49.9128i			0.892867i	0.894817	+	0.446434i	$0.147306\pi$
$5$			49.9128i			0.892867i	−0.894817	+	0.446434i	$0.852694\pi$
$6$		0			0
$7$	−258.614			−1.99484			−0.997418	−	0.0718079i	$-0.977123\pi$
$7$	−258.614			−1.99484			−0.997418	+	0.0718079i	$0.977123\pi$
$8$		0			0
$9$	−226.638	−	87.6587i	−0.932668	−	0.360736i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−91.5268	+	490.362i	−0.183483	+	0.983023i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	764.854	+	142.761i	0.877709	+	0.163826i
$16$	1024.00			1.00000
$17$		−	683.622i		−	0.573712i	−0.957974	−	0.286856i	$-0.907390\pi$
$17$		−	683.622i		−	0.573712i	0.957974	−	0.286856i	$-0.0926101\pi$
$18$		0			0
$19$	896.052			0.569442			0.284721	−	0.958610i	$-0.408099\pi$
$19$	896.052			0.569442			0.284721	+	0.958610i	$0.408099\pi$
$20$		−	1597.21i		−	0.892867i
$21$	−739.692	+	3962.96i	−0.366018	+	1.96097i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	633.714			0.202788
$26$		0			0
$27$	−1991.50	+	3222.24i	−0.525740	+	0.850645i
$28$	8275.66			1.99484
$29$			3246.37i			0.716808i	0.933567	+	0.358404i	$0.116679\pi$
$29$			3246.37i			0.716808i	−0.933567	+	0.358404i	$0.883321\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	12908.2i		−	1.78112i
$36$	7252.43	+	2805.08i	0.932668	+	0.360736i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	18900.1i		−	1.75592i	−0.478737	−	0.877958i	$-0.658905\pi$
$41$		−	18900.1i		−	1.75592i	0.478737	−	0.877958i	$-0.341095\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	4375.29	−	11312.2i	0.322089	−	0.832749i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	2928.86	−	15691.6i	0.183483	−	0.983023i
$49$	50074.3			2.97937
$50$		0			0
$51$	−10475.7	−	1955.30i	−0.563972	−	0.105266i
$52$		0			0
$53$			38850.3i			1.89978i	0.312578	+	0.949892i	$0.398808\pi$
$53$			38850.3i			1.89978i	−0.312578	+	0.949892i	$0.601192\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2562.90	−	13730.9i	0.104483	−	0.559774i
$58$		0			0
$59$		−	26738.1i		−	1.00000i
$59$		−	26738.1i		−	1.00000i
$60$	−24475.3	−	4568.36i	−0.877709	−	0.163826i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	58611.9	+	22669.8i	1.86052	+	0.719609i
$64$	−32768.0			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			21875.9i			0.573712i
$69$		0			0
$70$		0			0
$71$			59459.7i			1.39984i	0.714223	+	0.699918i	$0.246780\pi$
$71$			59459.7i			1.39984i	−0.714223	+	0.699918i	$0.753220\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1812.56	−	9710.91i	0.0372082	−	0.199346i
$76$	−28673.7			−0.569442
$77$		0			0
$78$		0			0
$79$	96849.2			1.74594			0.872968	−	0.487777i	$-0.162192\pi$
$79$	96849.2			1.74594			0.872968	+	0.487777i	$0.162192\pi$
$80$			51110.7i			0.892867i
$81$	43680.9	+	39733.7i	0.739740	+	0.672893i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	23670.1	−	126815.i	0.366018	−	1.96097i
$85$	34121.5			0.512248
$86$		0			0
$87$	49746.7	+	9285.31i	0.704639	+	0.131522i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			44724.5i			0.508436i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−20278.8			−0.202788
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−197802.	−	36920.1i	−1.75089	−	0.326806i
$106$		0			0
$107$		−	148039.i		−	1.25002i	−0.780618	−	0.625009i	$-0.785095\pi$
$107$		−	148039.i		−	1.25002i	0.780618	−	0.625009i	$-0.214905\pi$
$108$	63728.0	−	103112.i	0.525740	−	0.850645i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−264821.			−1.99484
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	103884.i		−	0.716808i
$117$		0			0
$118$		0			0
$119$			176794.i			1.14446i
$120$		0			0
$121$	−161051.			−1.00000
$122$		0			0
$123$	−289621.	−	54058.2i	−1.72611	−	0.322180i
$124$		0			0
$125$			187608.i			1.07393i
$126$		0			0
$127$	−169429.			−0.932135			−0.466067	−	0.884749i	$-0.654330\pi$
$127$	−169429.			−0.932135			−0.466067	+	0.884749i	$0.654330\pi$
$128$		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
$133$	−231732.			−1.13594
$134$		0			0
$135$	−160831.	−	99401.3i	−0.759513	−	0.469416i
$136$		0			0
$137$			426180.i			1.93995i	0.243195	+	0.969977i	$0.421804\pi$
$137$			426180.i			1.93995i	−0.243195	+	0.969977i	$0.578196\pi$
$138$		0			0
$139$	399275.			1.75281			0.876406	−	0.481574i	$-0.159935\pi$
$139$	399275.			1.75281			0.876406	+	0.481574i	$0.159935\pi$
$140$			413061.i			1.78112i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−232078.	−	89762.5i	−0.932668	−	0.360736i
$145$	−162035.			−0.640014
$146$		0			0
$147$	143223.	−	767330.i	0.546664	−	2.92879i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−59925.4	+	154935.i	−0.206958	+	0.535083i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	595334.	+	111120.i	1.86753	+	0.348578i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	537581.			1.58480			0.792401	−	0.610001i	$-0.208831\pi$
$163$	537581.			1.58480			0.792401	+	0.610001i	$0.208831\pi$
$164$			604802.i			1.75592i
$165$		0			0
$166$		0			0
$167$		−	423603.i		−	1.17535i	−0.809096	−	0.587676i	$-0.800043\pi$
$167$		−	423603.i		−	1.17535i	0.809096	−	0.587676i	$-0.199957\pi$
$168$		0			0
$169$	371293.			1.00000
$170$		0			0
$171$	−203080.	−	78546.8i	−0.531100	−	0.205418i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−163887.			−0.404530
$176$		0			0
$177$	−409729.	−	76476.6i	−0.983023	−	0.183483i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−140009.	+	361989.i	−0.322089	+	0.832749i
$181$	−843867.			−1.91460			−0.957299	−	0.289101i	$-0.906644\pi$
$181$	−843867.			−1.91460			−0.957299	+	0.289101i	$0.906644\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	515030.	−	833317.i	1.04877	−	1.69690i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−93723.5	+	502131.i	−0.183483	+	0.983023i
$193$	1.03055e6			1.99148			0.995739	−	0.0922178i	$-0.0293956\pi$
$193$	1.03055e6			1.99148			0.995739	+	0.0922178i	$0.0293956\pi$
$194$		0			0
$195$		0			0
$196$	−1.60238e6			−2.97937
$197$			1.07084e6i			1.96588i	0.183921	+	0.982941i	$0.441121\pi$
$197$			1.07084e6i			1.96588i	−0.183921	+	0.982941i	$0.558879\pi$
$198$		0			0
$199$	−82065.4			−0.146902			−0.0734510	−	0.997299i	$-0.523401\pi$
$199$	−82065.4			−0.146902			−0.0734510	+	0.997299i	$0.523401\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	839557.i		−	1.42992i
$204$	335222.	+	62569.8i	0.563972	+	0.105266i
$205$	943355.			1.56780
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	1.24321e6i		−	1.89978i
$213$	911150.	+	170068.i	1.37607	+	0.256846i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	447431.			0.602510			0.301255	−	0.953544i	$-0.402595\pi$
$223$	447431.			0.602510			0.301255	+	0.953544i	$0.402595\pi$
$224$		0			0
$225$	−143624.	−	55550.6i	−0.189134	−	0.0731530i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−82012.8	+	439390.i	−0.104483	+	0.559774i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		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
$236$			855618.i			1.00000i
$237$	277009.	−	1.48410e6i	0.320349	−	1.71630i
$238$		0			0
$239$		−	1.51927e6i		−	1.72044i	−0.509925	−	0.860219i	$-0.670327\pi$
$239$		−	1.51927e6i		−	1.72044i	0.509925	−	0.860219i	$-0.329673\pi$
$240$	783211.	+	146187.i	0.877709	+	0.163826i
$241$	1.79577e6			1.99163			0.995813	−	0.0914190i	$-0.0291403\pi$
$241$	1.79577e6			1.99163			0.995813	+	0.0914190i	$0.0291403\pi$
$242$		0			0
$243$	733808.	−	555711.i	0.797199	−	0.603717i
$244$		0			0
$245$			2.49935e6i			2.66019i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.85585e6i		−	1.85934i	−0.368392	−	0.929671i	$-0.620091\pi$
$251$		−	1.85585e6i		−	1.85934i	0.368392	−	0.929671i	$-0.379909\pi$
$252$	−1.87558e6	−	725434.i	−1.86052	−	0.719609i
$253$		0			0
$254$		0			0
$255$	97594.7	−	522871.i	0.0939887	−	0.503552i
$256$	1.04858e6			1.00000
$257$			1.35202e6i			1.27688i	0.769672	+	0.638440i	$0.220420\pi$
$257$			1.35202e6i			1.27688i	−0.769672	+	0.638440i	$0.779580\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	284573.	−	735752.i	0.258578	−	0.668544i
$262$		0			0
$263$			1.55233e6i			1.38387i	0.721959	+	0.691936i	$0.243242\pi$
$263$			1.55233e6i			1.38387i	−0.721959	+	0.691936i	$0.756758\pi$
$264$		0			0
$265$	−1.93912e6			−1.69625
$266$		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
$271$	137643.			0.113850			0.0569248	−	0.998378i	$-0.481870\pi$
$271$	137643.			0.113850			0.0569248	+	0.998378i	$0.481870\pi$
$272$		−	700029.i		−	0.573712i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	800959.			0.627207			0.313603	−	0.949554i	$-0.398464\pi$
$277$	800959.			0.627207			0.313603	+	0.949554i	$0.398464\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	125876.i		−	0.0950996i	−0.998869	−	0.0475498i	$-0.984859\pi$
$281$		−	125876.i		−	0.0950996i	0.998869	−	0.0475498i	$-0.0151413\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	1.90271e6i		−	1.39984i
$285$	685349.	+	127921.i	0.499804	+	0.0932892i
$286$		0			0
$287$			4.88783e6i			3.50277i
$288$		0			0
$289$	952518.			0.670855
$290$		0			0
$291$		0			0
$292$		0			0
$293$			2.68260e6i			1.82552i	0.408492	+	0.912762i	$0.366055\pi$
$293$			2.68260e6i			1.82552i	−0.408492	+	0.912762i	$0.633945\pi$
$294$		0			0
$295$	1.33457e6			0.892867
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−58001.8	+	310749.i	−0.0372082	+	0.199346i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	917557.			0.569442
$305$		0			0
$306$		0			0
$307$	1.49127e6			0.903045			0.451522	−	0.892260i	$-0.350881\pi$
$307$	1.49127e6			0.903045			0.451522	+	0.892260i	$0.350881\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	1.58709e6i		−	0.930469i	−0.885188	−	0.465234i	$-0.845970\pi$
$311$		−	1.58709e6i		−	0.930469i	0.885188	−	0.465234i	$-0.154030\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−1.13151e6	+	2.92548e6i	−0.642515	+	1.66120i
$316$	−3.09917e6			−1.74594
$317$		−	3.15415e6i		−	1.76293i	−0.472250	−	0.881465i	$-0.656558\pi$
$317$		−	3.15415e6i		−	1.76293i	0.472250	−	0.881465i	$-0.343442\pi$
$318$		0			0
$319$		0			0
$320$		−	1.63554e6i		−	0.892867i
$321$	−2.26852e6	−	423422.i	−1.22880	−	0.229357i
$322$		0			0
$323$		−	612561.i		−	0.326695i
$324$	−1.39779e6	−	1.27148e6i	−0.739740	−	0.672893i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.36568e6			−1.18682			−0.593411	−	0.804900i	$-0.702219\pi$
$331$	−2.36568e6			−1.18682			−0.593411	+	0.804900i	$0.702219\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−757445.	+	4.05807e6i	−0.366018	+	1.96097i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	−1.09189e6			−0.512248
$341$		0			0
$342$		0			0
$343$	−8.60341e6			−3.94853
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−1.59190e6	−	297130.i	−0.704639	−	0.131522i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−2.96780e6			−1.24987
$356$		0			0
$357$	2.70916e6	+	505670.i	1.12503	+	0.209989i
$358$		0			0
$359$		−	4.23515e6i		−	1.73434i	−0.498016	−	0.867168i	$-0.665938\pi$
$359$		−	4.23515e6i		−	1.73434i	0.498016	−	0.867168i	$-0.334062\pi$
$360$		0			0
$361$	−1.67319e6			−0.675736
$362$		0			0
$363$	−460640.	+	2.46792e6i	−0.183483	+	0.983023i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−1.65676e6	+	4.28348e6i	−0.633421	+	1.63769i
$370$		0			0
$371$		−	1.00472e7i		−	3.78976i
$372$		0			0
$373$	5.36607e6			1.99703			0.998514	−	0.0544950i	$-0.0173549\pi$
$373$	5.36607e6			1.99703			0.998514	+	0.0544950i	$0.0173549\pi$
$374$		0			0
$375$	2.87487e6	+	536599.i	1.05570	+	0.197048i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.83372e6			1.37095			0.685475	−	0.728096i	$-0.259594\pi$
$379$	3.83372e6			1.37095			0.685475	+	0.728096i	$0.259594\pi$
$380$		−	1.43118e6i		−	0.508436i
$381$	−484603.	+	2.59630e6i	−0.171031	+	0.916310i
$382$		0			0
$383$		−	5.18018e6i		−	1.80446i	−0.431252	−	0.902231i	$-0.641928\pi$
$383$		−	5.18018e6i		−	1.80446i	0.431252	−	0.902231i	$-0.358072\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.43319e6i		−	0.815273i	−0.913144	−	0.407636i	$-0.866353\pi$
$389$		−	2.43319e6i		−	0.815273i	0.913144	−	0.407636i	$-0.133647\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			4.83401e6i			1.55889i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−662803.	+	3.55102e6i	−0.208426	+	1.11666i
$400$	648923.			0.202788
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.98322e6	+	2.18023e6i	−0.600804	+	0.660489i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	6.53070e6	+	1.21897e6i	1.90702	+	0.355948i
$412$		0			0
$413$			6.91485e6i			1.99484i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.14201e6	−	6.11841e6i	0.321611	−	1.72305i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	6.32967e6	+	1.18144e6i	1.75089	+	0.326806i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	433221.i		−	0.116342i
$426$		0			0
$427$		0			0
$428$			4.73724e6i			1.25002i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−2.03930e6	+	3.29957e6i	−0.525740	+	0.850645i
$433$	−3.85408e6			−0.987874			−0.493937	−	0.869498i	$-0.664443\pi$
$433$	−3.85408e6			−0.987874			−0.493937	+	0.869498i	$0.664443\pi$
$434$		0			0
$435$	−463456.	+	2.48300e6i	−0.117432	+	0.629149i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−7.86257e6			−1.94717			−0.973584	−	0.228328i	$-0.926674\pi$
$439$	−7.86257e6			−1.94717			−0.973584	+	0.228328i	$0.926674\pi$
$440$		0			0
$441$	−1.13488e7	−	4.38945e6i	−2.77877	−	1.07477i
$442$		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
$448$	8.47427e6			1.99484
$449$			6.48642e6i			1.51841i	0.650851	+	0.759205i	$0.274412\pi$
$449$			6.48642e6i			1.51841i	−0.650851	+	0.759205i	$0.725588\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	2.20279e6	+	1.36143e6i	0.488025	+	0.301623i
$460$		0			0
$461$		−	6.82636e6i		−	1.49602i	−0.663689	−	0.748009i	$-0.731010\pi$
$461$		−	6.82636e6i		−	1.49602i	0.663689	−	0.748009i	$-0.268990\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			3.32428e6i			0.716808i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	567841.			0.115476
$476$		−	5.65742e6i		−	1.14446i
$477$	3.40557e6	−	8.80496e6i	0.685320	−	1.77187i
$478$		0			0
$479$			8.05733e6i			1.60455i	0.596957	+	0.802273i	$0.296376\pi$
$479$			8.05733e6i			1.60455i	−0.596957	+	0.802273i	$0.703624\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	5.15363e6			1.00000
$485$		0			0
$486$		0			0
$487$	8.12788e6			1.55294			0.776470	−	0.630154i	$-0.217008\pi$
$487$	8.12788e6			1.55294			0.776470	+	0.630154i	$0.217008\pi$
$488$		0			0
$489$	1.53760e6	−	8.23779e6i	0.290784	−	1.55790i
$490$		0			0
$491$		−	2.70444e6i		−	0.506259i	−0.967432	−	0.253130i	$-0.918540\pi$
$491$		−	2.70444e6i		−	0.506259i	0.967432	−	0.253130i	$-0.0814600\pi$
$492$	9.26788e6	+	1.72986e6i	1.72611	+	0.322180i
$493$	2.21929e6			0.411241
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1.53771e7i		−	2.79244i
$498$		0			0
$499$	3.42120e6			0.615073			0.307536	−	0.951536i	$-0.400495\pi$
$499$	3.42120e6			0.615073			0.307536	+	0.951536i	$0.400495\pi$
$500$		−	6.00345e6i		−	1.07393i
$501$	−6.49121e6	−	1.21160e6i	−1.15540	−	0.215657i
$502$		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
$507$	1.06198e6	−	5.68962e6i	0.183483	−	0.983023i
$508$	5.42173e6			0.932135
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.78449e6	+	2.88730e6i	−0.299378	+	0.484393i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.27107e6i			1.49636i	0.663497	+	0.748179i	$0.269072\pi$
$521$			9.27107e6i			1.49636i	−0.663497	+	0.748179i	$0.730928\pi$
$522$		0			0
$523$	1.24612e7			1.99208			0.996041	−	0.0888983i	$-0.0283346\pi$
$523$	1.24612e7			1.99208			0.996041	+	0.0888983i	$0.0283346\pi$
$524$		0			0
$525$	−468753.	+	2.51138e6i	−0.0742243	+	0.397662i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.43634e6			−1.00000
$530$		0			0
$531$	−2.34383e6	+	6.05987e6i	−0.360736	+	0.932668i
$532$	7.41542e6			1.13594
$533$		0			0
$534$		0			0
$535$	7.38903e6			1.11610
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	5.14659e6	+	3.18084e6i	0.759513	+	0.469416i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−2.41364e6	+	1.29313e7i	−0.351296	+	1.88209i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.06008e7			1.51485			0.757427	−	0.652920i	$-0.226456\pi$
$547$	1.06008e7			1.51485			0.757427	+	0.652920i	$0.226456\pi$
$548$		−	1.36378e7i		−	1.93995i
$549$		0			0
$550$		0			0
$551$			2.90892e6i			0.408180i
$552$		0			0
$553$	−2.50466e7			−3.48286
$554$		0			0
$555$		0			0
$556$	−1.27768e7			−1.75281
$557$			1.46439e7i			1.99994i	0.00752886	+	0.999972i	$0.497603\pi$
$557$			1.46439e7i			1.99994i	−0.00752886	+	0.999972i	$0.502397\pi$
$558$		0			0
$559$		0			0
$560$		−	1.32180e7i		−	1.78112i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.12965e7	−	1.02757e7i	−1.47566	−	1.34231i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	7.42649e6	+	2.87240e6i	0.932668	+	0.360736i
$577$	−8.86441e6			−1.10844			−0.554218	−	0.832372i	$-0.686982\pi$
$577$	−8.86441e6			−1.10844			−0.554218	+	0.832372i	$0.686982\pi$
$578$		0			0
$579$	2.94759e6	−	1.57919e7i	0.365402	−	1.95767i
$580$	5.18513e6			0.640014
$581$		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	$0.5\pi$
$588$	−4.58315e6	+	2.45546e7i	−0.546664	+	2.92879i
$589$		0			0
$590$		0			0
$591$	1.64093e7	+	3.06282e6i	1.93251	+	0.360706i
$592$		0			0
$593$			1.69832e7i			1.98327i	0.129057	+	0.991637i	$0.458805\pi$
$593$			1.69832e7i			1.98327i	−0.129057	+	0.991637i	$0.541195\pi$
$594$		0			0
$595$	−8.82430e6			−1.02185
$596$		0			0
$597$	−234725.	+	1.25756e6i	−0.0269540	+	0.144408i
$598$		0			0
$599$			9.88518e6i			1.12569i	0.826564	+	0.562843i	$0.190293\pi$
$599$			9.88518e6i			1.12569i	−0.826564	+	0.562843i	$0.809707\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	8.03850e6i		−	0.892867i
$606$		0			0
$607$	1.07713e7			1.18657			0.593287	−	0.804991i	$-0.297830\pi$
$607$	1.07713e7			1.18657			0.593287	+	0.804991i	$0.297830\pi$
$608$		0			0
$609$	−1.28652e7	−	2.40131e6i	−1.40564	−	0.262365i
$610$		0			0
$611$		0			0
$612$	1.91761e6	−	4.95792e6i	0.206958	−	0.535083i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	2.69820e6	−	1.44558e7i	0.287664	−	1.54118i
$616$		0			0
$617$			1.51272e7i			1.59973i	0.600180	+	0.799865i	$0.295096\pi$
$617$			1.51272e7i			1.59973i	−0.600180	+	0.799865i	$0.704904\pi$
$618$		0			0
$619$	−1.31704e7			−1.38157			−0.690785	−	0.723060i	$-0.742735\pi$
$619$	−1.31704e7			−1.38157			−0.690785	+	0.723060i	$0.742735\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.38368e6			−0.756088
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.77761e7			1.77731			0.888655	−	0.458577i	$-0.151641\pi$
$631$	1.77761e7			1.77731			0.888655	+	0.458577i	$0.151641\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	8.45668e6i		−	0.832273i
$636$	−1.90507e7	−	3.55584e6i	−1.86753	−	0.348578i
$637$		0			0
$638$		0			0
$639$	5.21217e6	−	1.34759e7i	0.504971	−	1.30558i
$640$		0			0
$641$			1.91009e7i			1.83615i	0.396404	+	0.918076i	$0.370258\pi$
$641$			1.91009e7i			1.83615i	−0.396404	+	0.918076i	$0.629742\pi$
$642$		0			0
$643$	−1.83283e7			−1.74822			−0.874108	−	0.485733i	$-0.838553\pi$
$643$	−1.83283e7			−1.74822			−0.874108	+	0.485733i	$0.838553\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.12667e7i			1.99728i	0.0520897	+	0.998642i	$0.483412\pi$
$647$			2.12667e7i			1.99728i	−0.0520897	+	0.998642i	$0.516588\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.72026e7			−1.58480
$653$		−	7.59642e6i		−	0.697150i	−0.937281	−	0.348575i	$-0.886666\pi$
$653$		−	7.59642e6i		−	0.697150i	0.937281	−	0.348575i	$-0.113334\pi$
$654$		0			0
$655$		0			0
$656$		−	1.93537e7i		−	1.75592i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.91517e6			0.170492			0.0852460	−	0.996360i	$-0.472832\pi$
$661$	1.91517e6			0.170492			0.0852460	+	0.996360i	$0.472832\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.15664e7i		−	1.01425i
$666$		0			0
$667$		0			0
$668$			1.35553e7i			1.17535i
$669$	1.27975e6	−	6.85635e6i	0.110550	−	0.592281i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−1.26204e6	+	2.04198e6i	−0.106614	+	0.172501i
$676$	−1.18814e7			−1.00000
$677$		−	388942.i		−	0.0326147i	−0.999867	−	0.0163074i	$-0.994809\pi$
$677$		−	388942.i		−	0.0326147i	0.999867	−	0.0163074i	$-0.00519102\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	6.49855e6	+	2.51350e6i	0.531100	+	0.205418i
$685$	−2.12718e7			−1.73212
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.99289e7i			1.56503i
$696$		0			0
$697$	−1.29205e7			−1.00739
$698$		0			0
$699$		0			0
$700$	5.24440e6			0.404530
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	1.31113e7	+	2.44725e6i	0.983023	+	0.183483i
$709$	1.67805e7			1.25369			0.626845	−	0.779144i	$-0.284346\pi$
$709$	1.67805e7			1.25369			0.626845	+	0.779144i	$0.284346\pi$
$710$		0			0
$711$	−2.19497e7	−	8.48968e6i	−1.62838	−	0.629821i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−2.32809e7	−	4.34542e6i	−1.69123	−	0.315671i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	4.48030e6	−	1.15836e7i	0.322089	−	0.832749i
$721$		0			0
$722$		0			0
$723$	5.13628e6	−	2.75180e7i	0.365429	−	1.95781i
$724$	2.70037e7			1.91460
$725$			2.05727e6i			0.145360i
$726$		0			0
$727$	−1.14089e7			−0.800586			−0.400293	−	0.916387i	$-0.631092\pi$
$727$	−1.14089e7			−0.800586			−0.400293	+	0.916387i	$0.631092\pi$
$728$		0			0
$729$	−6.41676e6	−	1.28342e7i	−0.447195	−	0.894436i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.70725e7			1.17365			0.586824	−	0.809714i	$-0.300378\pi$
$733$	1.70725e7			1.17365			0.586824	+	0.809714i	$0.300378\pi$
$734$		0			0
$735$	3.82996e7	+	7.14867e6i	2.61502	+	0.488098i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.03276e7i			1.35087i	0.737419	+	0.675436i	$0.236044\pi$
$743$			2.03276e7i			1.35087i	−0.737419	+	0.675436i	$0.763956\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			3.82849e7i			2.49358i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−2.84387e7	−	5.30813e6i	−1.82778	−	0.341157i
$754$		0			0
$755$		0			0
$756$	−1.64810e7	+	2.66662e7i	−1.04877	+	1.69690i
$757$	6.84658e6			0.434244			0.217122	−	0.976144i	$-0.430333\pi$
$757$	6.84658e6			0.434244			0.217122	+	0.976144i	$0.430333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3.19363e7i		−	1.99905i	−0.0308916	−	0.999523i	$-0.509835\pi$
$761$		−	3.19363e7i		−	1.99905i	0.0308916	−	0.999523i	$-0.490165\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−7.73324e6	−	2.99105e6i	−0.477758	−	0.184786i
$766$		0			0
$767$		0			0
$768$	2.99915e6	−	1.60682e7i	0.183483	−	0.983023i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	2.07181e7	+	3.86706e6i	1.25520	+	0.234285i
$772$	−3.29776e7			−1.99148
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.69354e7i		−	0.999892i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.04606e7	−	6.46514e6i	−0.609749	−	0.376855i
$784$	5.12761e7			2.97937
$785$		0			0
$786$		0			0
$787$	3.71139e6			0.213599			0.106800	−	0.994281i	$-0.465940\pi$
$787$	3.71139e6			0.213599			0.106800	+	0.994281i	$0.465940\pi$
$788$		−	3.42668e7i		−	1.96588i
$789$	2.37877e7	+	4.44001e6i	1.36038	+	0.253917i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−5.54631e6	+	2.97148e7i	−0.311234	+	1.66746i
$796$	2.62609e6			0.146902
$797$		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			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$			2.68658e7i			1.42992i
$813$	393689.	−	2.10922e6i	0.0208894	−	0.111917i
$814$		0			0
$815$			2.68322e7i			1.41502i
$816$	−1.07271e7	−	2.00223e6i	−0.563972	−	0.105266i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−3.01874e7			−1.56780
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.44196e7i			1.24158i	0.783977	+	0.620790i	$0.213188\pi$
$827$			2.44196e7i			1.24158i	−0.783977	+	0.620790i	$0.786812\pi$
$828$		0			0
$829$	3.42457e7			1.73069			0.865345	−	0.501176i	$-0.167099\pi$
$829$	3.42457e7			1.73069			0.865345	+	0.501176i	$0.167099\pi$
$830$		0			0
$831$	2.29091e6	−	1.22737e7i	0.115082	−	0.616559i
$832$		0			0
$833$		−	3.42319e7i		−	1.70930i
$834$		0			0
$835$	2.11432e7			1.04943
$836$		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
$841$	9.97224e6			0.486186
$842$		0			0
$843$	−1.92891e6	−	360033.i	−0.0934850	−	0.0174491i
$844$		0			0
$845$			1.85323e7i			0.892867i
$846$		0			0
$847$	4.16501e7			1.99484
$848$			3.97827e7i			1.89978i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.91568e7	−	5.44216e6i	−1.37607	−	0.256846i
$853$	−2.32074e7			−1.09208			−0.546038	−	0.837760i	$-0.683865\pi$
$853$	−2.32074e7			−1.09208			−0.546038	+	0.837760i	$0.683865\pi$
$854$		0			0
$855$	3.92049e6	−	1.01363e7i	0.183411	−	0.474202i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$	7.49002e7	+	1.39802e7i	3.44330	+	0.642697i
$862$		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
$867$	2.72440e6	−	1.45962e7i	0.123090	−	0.659466i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	4.85181e7i		−	2.14232i
$876$		0			0
$877$	−3.58175e7			−1.57252			−0.786261	−	0.617895i	$-0.787986\pi$
$877$	−3.58175e7			−1.57252			−0.786261	+	0.617895i	$0.787986\pi$
$878$		0			0
$879$	4.11077e7	+	7.67282e6i	1.79453	+	0.334952i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	2.33740e7			1.00886			0.504430	−	0.863453i	$-0.331703\pi$
$883$	2.33740e7			1.00886			0.504430	+	0.863453i	$0.331703\pi$
$884$		0			0
$885$	3.81716e6	−	2.04507e7i	0.163826	−	0.877709i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	4.38168e7			1.85946
$890$		0			0
$891$		0			0
$892$	−1.43178e7			−0.602510
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	4.59596e6	+	1.77762e6i	0.189134	+	0.0731530i
$901$	2.65589e7			1.08993
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	4.21197e7i		−	1.70948i
$906$		0			0
$907$	−4.91346e7			−1.98321			−0.991606	−	0.129300i	$-0.958727\pi$
$907$	−4.91346e7			−1.98321			−0.991606	+	0.129300i	$0.958727\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.85382e7i			1.93771i	0.247638	+	0.968853i	$0.420346\pi$
$911$			4.85382e7i			1.93771i	−0.247638	+	0.968853i	$0.579654\pi$
$912$	2.62441e6	−	1.40605e7i	0.104483	−	0.559774i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	4.26534e6	−	2.28519e7i	0.165693	−	0.887714i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	4.48692e7			1.69658
$932$		0			0
$933$	−2.43203e7	−	4.53943e6i	−0.914672	−	0.170725i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		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
$944$		−	2.73798e7i		−	1.00000i
$945$	4.15932e7	+	2.57066e7i	1.51511	+	0.936408i
$946$		0			0
$947$		−	2.61702e6i		−	0.0948268i	−0.998875	−	0.0474134i	$-0.984902\pi$
$947$		−	2.61702e6i		−	0.0948268i	0.998875	−	0.0474134i	$-0.0150978\pi$
$948$	−8.86430e6	+	4.74912e7i	−0.320349	+	1.71630i
$949$		0			0
$950$		0			0
$951$	−4.83337e7	−	9.02156e6i	−1.73300	−	0.323467i
$952$		0			0
$953$		−	4.95637e7i		−	1.76779i	−0.467681	−	0.883897i	$-0.654911\pi$
$953$		−	4.95637e7i		−	1.76779i	0.467681	−	0.883897i	$-0.345089\pi$
$954$		0			0
$955$		0			0
$956$			4.86165e7i			1.72044i
$957$		0			0
$958$		0			0
$959$		−	1.10216e8i		−	3.86989i
$960$	−2.50627e7	−	4.67800e6i	−0.877709	−	0.163826i
$961$	2.86292e7			1.00000
$962$		0			0
$963$	−1.29769e7	+	3.35513e7i	−0.450926	+	1.16585i
$964$	−5.74646e7			−1.99163
$965$			5.14376e7i			1.77812i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−9.38677e6	−	1.75205e6i	−0.321149	−	0.0599430i
$970$		0			0
$971$		−	4.97290e7i		−	1.69263i	−0.532683	−	0.846315i	$-0.678816\pi$
$971$		−	4.97290e7i		−	1.69263i	0.532683	−	0.846315i	$-0.321184\pi$
$972$	−2.34819e7	+	1.77828e7i	−0.797199	+	0.603717i
$973$	−1.03258e8			−3.49657
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		−	7.99792e7i		−	2.66019i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−5.34484e7			−1.75527
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−6.76634e6	+	3.62512e7i	−0.217761	+	1.16667i
$994$		0			0
$995$		−	4.09611e6i		−	0.131164i
$996$		0			0
$997$	−4.65711e7			−1.48381			−0.741906	−	0.670504i	$-0.766078\pi$
$997$	−4.65711e7			−1.48381			−0.741906	+	0.670504i	$0.766078\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.6.d.a.176.3	✓	6
3.2	odd	2	inner	177.6.d.a.176.4	yes	6
59.58	odd	2	CM	177.6.d.a.176.3	✓	6
177.176	even	2	inner	177.6.d.a.176.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.d.a.176.3	✓	6	1.1	even	1	trivial
177.6.d.a.176.3	✓	6	59.58	odd	2	CM
177.6.d.a.176.4	yes	6	3.2	odd	2	inner
177.6.d.a.176.4	yes	6	177.176	even	2	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +O(q^{10})\)	\(q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +(-91.5268 + 490.362i) q^{12} +(764.854 + 142.761i) q^{15} +1024.00 q^{16} -683.622i q^{17} +896.052 q^{19} -1597.21i q^{20} +(-739.692 + 3962.96i) q^{21} +633.714 q^{25} +(-1991.50 + 3222.24i) q^{27} +8275.66 q^{28} +3246.37i q^{29} -12908.2i q^{35} +(7252.43 + 2805.08i) q^{36} -18900.1i q^{41} +(4375.29 - 11312.2i) q^{45} +(2928.86 - 15691.6i) q^{48} +50074.3 q^{49} +(-10475.7 - 1955.30i) q^{51} +38850.3i q^{53} +(2562.90 - 13730.9i) q^{57} -26738.1i q^{59} +(-24475.3 - 4568.36i) q^{60} +(58611.9 + 22669.8i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(1812.56 - 9710.91i) q^{75} -28673.7 q^{76} +96849.2 q^{79} +51110.7i q^{80} +(43680.9 + 39733.7i) q^{81} +(23670.1 - 126815. i) q^{84} +34121.5 q^{85} +(49746.7 + 9285.31i) q^{87} +44724.5i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 192 q^{4}+O(q^{10}) \) 6 * q - 192 * q^4	\( 6 q - 192 q^{4} + 3153 q^{15} + 6144 q^{16} - 3075 q^{21} - 18750 q^{25} - 11949 q^{27} + 81147 q^{45} + 100842 q^{49} - 131001 q^{57} - 100896 q^{60} + 180732 q^{63} - 196608 q^{64} - 197304 q^{75} + 98400 q^{84} + 221799 q^{87}+O(q^{100}) \) 6 * q - 192 * q^4 + 3153 * q^15 + 6144 * q^16 - 3075 * q^21 - 18750 * q^25 - 11949 * q^27 + 81147 * q^45 + 100842 * q^49 - 131001 * q^57 - 100896 * q^60 + 180732 * q^63 - 196608 * q^64 - 197304 * q^75 + 98400 * q^84 + 221799 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more