LMFDB - Embedded newform 177.6.d.a.176.6

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			176.6
Root			$-1.24353 - 1.20567i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.6.d.a.176.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +O(q^{10})$	$q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +(-378.902 - 324.446i) q^{12} +(1131.47 - 1321.38i) q^{15} +1024.00 q^{16} -683.622i q^{17} -3060.71 q^{19} +3571.09i q^{20} +(1340.17 + 1147.55i) q^{21} -9328.81 q^{25} +(-1991.50 + 3222.24i) q^{27} -3621.85 q^{28} -8946.38i q^{29} -12630.8i q^{35} +(-1196.94 - 7683.33i) q^{36} +524.845i q^{41} +(26794.8 - 4174.22i) q^{45} +(12124.9 + 10382.3i) q^{48} -3996.62 q^{49} +(6931.19 - 8094.56i) q^{51} -30496.7i q^{53} +(-36241.0 - 31032.3i) q^{57} -26738.1i q^{59} +(-36207.0 + 42284.3i) q^{60} +(4233.55 + 27175.7i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(-110460. - 94584.1i) q^{75} +97942.7 q^{76} -95289.8 q^{79} -114275. i q^{80} +(-56250.8 + 17961.9i) q^{81} +(-42885.3 - 36721.7i) q^{84} -76289.9 q^{85} +(90706.7 - 105931. i) q^{87} +341565. i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.5\pi$
$3$	11.8407	+	10.1389i	0.759581	+	0.650412i
$3$	11.8407	+	10.1389i	0.759581	+	0.650412i
$4$	−32.0000			−1.00000
$5$		−	111.597i		−	1.99630i	−0.0607856	−	0.998151i	$-0.519361\pi$
$5$		−	111.597i		−	1.99630i	0.0607856	−	0.998151i	$-0.480639\pi$
$6$		0			0
$7$	113.183			0.873044			0.436522	−	0.899694i	$-0.356210\pi$
$7$	113.183			0.873044			0.436522	+	0.899694i	$0.356210\pi$
$8$		0			0
$9$	37.4045	+	240.104i	0.153928	+	0.988082i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−378.902	−	324.446i	−0.759581	−	0.650412i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	1131.47	−	1321.38i	1.29842	−	1.51635i
$16$	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
$19$	−3060.71			−1.94508			−0.972541	−	0.232730i	$-0.925234\pi$
$19$	−3060.71			−1.94508			−0.972541	+	0.232730i	$0.925234\pi$
$20$			3571.09i			1.99630i
$21$	1340.17	+	1147.55i	0.663148	+	0.567838i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−9328.81			−2.98522
$26$		0			0
$27$	−1991.50	+	3222.24i	−0.525740	+	0.850645i
$28$	−3621.85			−0.873044
$29$		−	8946.38i		−	1.97539i	−0.156396	−	0.987694i	$-0.549988\pi$
$29$		−	8946.38i		−	1.97539i	0.156396	−	0.987694i	$-0.450012\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	12630.8i		−	1.74286i
$36$	−1196.94	−	7683.33i	−0.153928	−	0.988082i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			524.845i			0.0487609i	0.999703	+	0.0243804i	$0.00776130\pi$
$41$			524.845i			0.0487609i	−0.999703	+	0.0243804i	$0.992239\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	26794.8	−	4174.22i	1.97251	−	0.307287i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	12124.9	+	10382.3i	0.759581	+	0.650412i
$49$	−3996.62			−0.237795
$50$		0			0
$51$	6931.19	−	8094.56i	0.373149	−	0.435781i
$52$		0			0
$53$		−	30496.7i		−	1.49129i	−0.666342	−	0.745646i	$-0.732141\pi$
$53$		−	30496.7i		−	1.49129i	0.666342	−	0.745646i	$-0.267859\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−36241.0	−	31032.3i	−1.47745	−	1.26511i
$58$		0			0
$59$		−	26738.1i		−	1.00000i
$59$		−	26738.1i		−	1.00000i
$60$	−36207.0	+	42284.3i	−1.29842	+	1.51635i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	4233.55	+	27175.7i	0.134386	+	0.862639i
$64$	−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
$75$	−110460.	−	94584.1i	−2.26752	−	1.94162i
$76$	97942.7			1.94508
$77$		0			0
$78$		0			0
$79$	−95289.8			−1.71782			−0.858912	−	0.512124i	$-0.828859\pi$
$79$	−95289.8			−1.71782			−0.858912	+	0.512124i	$0.828859\pi$
$80$		−	114275.i		−	1.99630i
$81$	−56250.8	+	17961.9i	−0.952612	+	0.304187i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−42885.3	−	36721.7i	−0.663148	−	0.567838i
$85$	−76289.9			−1.14530
$86$		0			0
$87$	90706.7	−	105931.i	1.28482	−	1.50047i
$88$		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
$95$			341565.i			3.88297i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	298522.			2.98522
$101$		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
$105$	128063.	−	149558.i	1.13358	−	1.32384i
$106$		0			0
$107$		−	86105.4i		−	0.727061i	−0.931582	−	0.363531i	$-0.881571\pi$
$107$		−	86105.4i		−	0.727061i	0.931582	−	0.363531i	$-0.118429\pi$
$108$	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
$112$	115899.			0.873044
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			286284.i			1.97539i
$117$		0			0
$118$		0			0
$119$		−	77374.4i		−	0.500875i
$120$		0			0
$121$	−161051.			−1.00000
$122$		0			0
$123$	−5321.36	+	6214.53i	−0.0317147	+	0.0370379i
$124$		0			0
$125$			692325.i			3.96310i
$126$		0			0
$127$	363256.			1.99850			0.999249	−	0.0387483i	$-0.0123371\pi$
$127$	363256.			1.99850			0.999249	+	0.0387483i	$0.0123371\pi$
$128$		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
$133$	−346420.			−1.69814
$134$		0			0
$135$	359591.	+	222245.i	1.69814	+	1.04954i
$136$		0			0
$137$		−	305627.i		−	1.39120i	−0.718427	−	0.695602i	$-0.755138\pi$
$137$		−	305627.i		−	1.39120i	0.718427	−	0.695602i	$-0.244862\pi$
$138$		0			0
$139$	399275.			1.75281			0.876406	−	0.481574i	$-0.159935\pi$
$139$	399275.			1.75281			0.876406	+	0.481574i	$0.159935\pi$
$140$			404187.i			1.74286i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	38302.2	+	245866.i	0.153928	+	0.988082i
$145$	−998387.			−3.94347
$146$		0			0
$147$	−47322.8	−	40521.4i	−0.180625	−	0.154665i
$148$		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
$153$	164140.	−	25570.5i	0.566874	−	0.0883102i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	309203.	−	361102.i	0.969955	−	1.13276i
$160$		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	$0.208831\pi$
$164$		−	16795.0i		−	0.0487609i
$165$		0			0
$166$		0			0
$167$		−	293269.i		−	0.813720i	−0.913491	−	0.406860i	$-0.866624\pi$
$167$		−	293269.i		−	0.813720i	0.913491	−	0.406860i	$-0.133376\pi$
$168$		0			0
$169$	371293.			1.00000
$170$		0			0
$171$	−114484.	−	734888.i	−0.299403	−	1.92190i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−1.05586e6			−2.60623
$176$		0			0
$177$	271095.	−	316598.i	0.650412	−	0.759581i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−857434.	+	133575.i	−1.97251	+	0.307287i
$181$	642636.			1.45804			0.729018	−	0.684494i	$-0.239977\pi$
$181$	642636.			1.45804			0.729018	+	0.684494i	$0.239977\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−225404.	+	364703.i	−0.458994	+	0.742651i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−387996.	−	332232.i	−0.759581	−	0.650412i
$193$	−597929.			−1.15546			−0.577732	−	0.816226i	$-0.696062\pi$
$193$	−597929.			−1.15546			−0.577732	+	0.816226i	$0.696062\pi$
$194$		0			0
$195$		0			0
$196$	127892.			0.237795
$197$			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
$199$	1.00601e6			1.80082			0.900412	−	0.435039i	$-0.143265\pi$
$199$	1.00601e6			1.80082			0.900412	+	0.435039i	$0.143265\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.01258e6i		−	1.72460i
$204$	−221798.	+	259026.i	−0.373149	+	0.435781i
$205$	58570.9			0.0973414
$206$		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			$\pi$
$212$			975894.i			1.49129i
$213$	−602858.	+	704045.i	−0.910471	+	1.06329i
$214$		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
$225$	−348939.	−	2.23989e6i	−0.459509	−	2.94964i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	1.15971e6	+	993034.i	1.47745	+	1.26511i
$229$		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
$237$	−1.12830e6	−	966136.i	−1.30483	−	1.11729i
$238$		0			0
$239$		−	20308.4i		−	0.0229975i	−0.999934	−	0.0114987i	$-0.996340\pi$
$239$		−	20308.4i		−	0.0229975i	0.999934	−	0.0114987i	$-0.00366024\pi$
$240$	1.15863e6	−	1.35310e6i	1.29842	−	1.51635i
$241$	−1.04065e6			−1.15415			−0.577077	−	0.816689i	$-0.695807\pi$
$241$	−1.04065e6			−1.15415			−0.577077	+	0.816689i	$0.695807\pi$
$242$		0			0
$243$	−848164.	−	357641.i	−0.921434	−	0.388536i
$244$		0			0
$245$			446009.i			0.474710i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			291051.i			0.291598i	0.989314	+	0.145799i	$0.0465753\pi$
$251$			291051.i			0.291598i	−0.989314	+	0.145799i	$0.953425\pi$
$252$	−135474.	−	869622.i	−0.134386	−	0.862639i
$253$		0			0
$254$		0			0
$255$	−903326.	−	773498.i	−0.869950	−	0.744918i
$256$	1.04858e6			1.00000
$257$		−	2.08757e6i		−	1.97155i	−0.168069	−	0.985775i	$-0.553753\pi$
$257$		−	2.08757e6i		−	1.97155i	0.168069	−	0.985775i	$-0.446247\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.14806e6	−	334635.i	1.95185	−	0.304067i
$262$		0			0
$263$		−	2.17886e6i		−	1.94241i	−0.238255	−	0.971203i	$-0.576575\pi$
$263$		−	2.17886e6i		−	1.94241i	0.238255	−	0.971203i	$-0.423425\pi$
$264$		0			0
$265$	−3.40333e6			−2.97707
$266$		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
$271$	2.02182e6			1.67232			0.836159	−	0.548488i	$-0.184796\pi$
$271$	2.02182e6			1.67232			0.836159	+	0.548488i	$0.184796\pi$
$272$		−	700029.i		−	0.573712i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50077e6			−1.95828			−0.979140	−	0.203189i	$-0.934870\pi$
$277$	−2.50077e6			−1.95828			−0.979140	+	0.203189i	$0.934870\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.35294e6i			1.77764i	0.458255	+	0.888821i	$0.348475\pi$
$281$			2.35294e6i			1.77764i	−0.458255	+	0.888821i	$0.651525\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	1.90271e6i		−	1.39984i
$285$	−3.46310e6	+	4.04437e6i	−2.52553	+	2.94943i
$286$		0			0
$287$			59403.5i			0.0425704i
$288$		0			0
$289$	952518.			0.670855
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	301590.i		−	0.205234i	−0.994721	−	0.102617i	$-0.967278\pi$
$293$		−	301590.i		−	0.205234i	0.994721	−	0.102617i	$-0.0327215\pi$
$294$		0			0
$295$	−2.98388e6			−1.99630
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	3.53471e6	+	3.02669e6i	2.26752	+	1.94162i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−3.13417e6			−1.94508
$305$		0			0
$306$		0			0
$307$	1.80647e6			1.09392			0.546959	−	0.837160i	$-0.315786\pi$
$307$	1.80647e6			1.09392			0.546959	+	0.837160i	$0.315786\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.40870e6i			1.99842i	0.0396890	+	0.999212i	$0.487363\pi$
$311$			3.40870e6i			1.99842i	−0.0396890	+	0.999212i	$0.512637\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	3.03271e6	−	472450.i	1.72209	−	0.268275i
$316$	3.04927e6			1.71782
$317$		−	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
$320$			3.65680e6i			1.99630i
$321$	873016.	−	1.01955e6i	0.472890	−	0.552262i
$322$		0			0
$323$			2.09237e6i			1.11592i
$324$	1.80003e6	−	574782.i	0.952612	−	0.304187i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.59606e6			−0.800717			−0.400358	−	0.916359i	$-0.631114\pi$
$331$	−1.59606e6			−0.800717			−0.400358	+	0.916359i	$0.631114\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	1.37233e6	+	1.17509e6i	0.663148	+	0.567838i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	2.44128e6			1.14530
$341$		0			0
$342$		0			0
$343$	−2.35462e6			−1.08065
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−2.90261e6	+	3.38981e6i	−1.28482	+	1.50047i
$349$		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
$355$	6.63551e6			2.79450
$356$		0			0
$357$	784493.	−	916167.i	0.325775	−	0.380456i
$358$		0			0
$359$			11181.8i			0.00457905i	0.999997	+	0.00228952i	$0.000728778\pi$
$359$			11181.8i			0.00457905i	−0.999997	+	0.00228952i	$0.999271\pi$
$360$		0			0
$361$	6.89184e6			2.78335
$362$		0			0
$363$	−1.90696e6	−	1.63288e6i	−0.759581	−	0.650412i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−126017.	+	19631.6i	−0.0481797	+	0.00750566i
$370$		0			0
$371$		−	3.45170e6i		−	1.30196i
$372$		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
$375$	−7.01943e6	+	8.19761e6i	−2.57765	+	3.01030i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.60967e6			0.575624			0.287812	−	0.957687i	$-0.407072\pi$
$379$	1.60967e6			0.575624			0.287812	+	0.957687i	$0.407072\pi$
$380$		−	1.09301e7i		−	3.88297i
$381$	4.30121e6	+	3.68303e6i	1.51802	+	1.29985i
$382$		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
$395$			1.06340e7i			3.42929i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−4.10186e6	−	3.51233e6i	−1.28988	−	1.10449i
$400$	−9.55271e6			−2.98522
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.00449e6	+	6.27740e6i	0.607249	+	1.90170i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	3.09873e6	−	3.61884e6i	0.904856	−	1.05673i
$412$		0			0
$413$		−	3.02629e6i		−	0.873044i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.72770e6	+	4.04822e6i	1.33140	+	1.14005i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−4.09802e6	+	4.78586e6i	−1.13358	+	1.32384i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			6.37738e6i			1.71266i
$426$		0			0
$427$		0			0
$428$			2.75537e6i			0.727061i
$429$		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
$433$	−3.94851e6			−1.01208			−0.506039	−	0.862511i	$-0.668891\pi$
$433$	−3.94851e6			−1.01208			−0.506039	+	0.862511i	$0.668891\pi$
$434$		0			0
$435$	−1.18216e7	−	1.01226e7i	−2.99539	−	2.56488i
$436$		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
$441$	−149491.	−	959604.i	−0.0366033	−	0.234961i
$442$		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
$448$	−3.70878e6			−0.873044
$449$		−	8.05890e6i		−	1.88651i	−0.332065	−	0.943256i	$-0.607745\pi$
$449$		−	8.05890e6i		−	1.88651i	0.332065	−	0.943256i	$-0.392255\pi$
$450$		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			$\pi$
$464$		−	9.16110e6i		−	1.97539i
$465$		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
$475$	2.85528e7			5.80650
$476$			2.47598e6i			0.500875i
$477$	7.32237e6	−	1.14071e6i	1.47352	−	0.229552i
$478$		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
$487$	−9.77648e6			−1.86793			−0.933964	−	0.357366i	$-0.883675\pi$
$487$	−9.77648e6			−1.86793			−0.933964	+	0.357366i	$0.883675\pi$
$488$		0			0
$489$	6.36534e6	+	5.45049e6i	1.20379	+	1.03077i
$490$		0			0
$491$		−	7.59906e6i		−	1.42251i	−0.702933	−	0.711256i	$-0.748127\pi$
$491$		−	7.59906e6i		−	1.42251i	0.702933	−	0.711256i	$-0.251873\pi$
$492$	170284.	−	198865.i	0.0317147	−	0.0370379i
$493$	−6.11595e6			−1.13330
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.72983e6i			1.22212i
$498$		0			0
$499$	−1.08778e7			−1.95565			−0.977823	−	0.209434i	$-0.932838\pi$
$499$	−1.08778e7			−1.95565			−0.977823	+	0.209434i	$0.932838\pi$
$500$		−	2.21544e7i		−	3.96310i
$501$	2.97343e6	−	3.47251e6i	0.529253	−	0.618087i
$502$		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
$507$	4.39637e6	+	3.76451e6i	0.759581	+	0.650412i
$508$	−1.16242e7			−1.99850
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.09540e6	−	9.86234e6i	1.02261	−	1.65458i
$514$		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
$523$	−5.26744e6			−0.842064			−0.421032	−	0.907046i	$-0.638332\pi$
$523$	−5.26744e6			−0.842064			−0.421032	+	0.907046i	$0.638332\pi$
$524$		0			0
$525$	−1.25022e7	−	1.07053e7i	−1.97964	−	1.69512i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.43634e6			−1.00000
$530$		0			0
$531$	6.41992e6	−	1.00012e6i	0.988082	−	0.153928i
$532$	1.10854e7			1.69814
$533$		0			0
$534$		0			0
$535$	−9.60908e6			−1.45143
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−1.15069e7	−	7.11183e6i	−1.69814	−	1.04954i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	7.60926e6	+	6.51563e6i	1.10750	+	0.948325i
$544$		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	$0.226456\pi$
$548$			9.78008e6i			1.39120i
$549$		0			0
$550$		0			0
$551$			2.73823e7i			3.84229i
$552$		0			0
$553$	−1.07852e7			−1.49973
$554$		0			0
$555$		0			0
$556$	−1.27768e7			−1.75281
$557$		−	7.41742e6i		−	1.01301i	−0.862236	−	0.506506i	$-0.830937\pi$
$557$		−	7.41742e6i		−	1.01301i	0.862236	−	0.506506i	$-0.169063\pi$
$558$		0			0
$559$		0			0
$560$		−	1.29340e7i		−	1.74286i
$561$		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
$567$	−6.36663e6	+	2.03298e6i	−0.831672	+	0.265568i
$568$		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
$576$	−1.22567e6	−	7.86773e6i	−0.153928	−	0.988082i
$577$	−7.09748e6			−0.887493			−0.443746	−	0.896152i	$-0.646351\pi$
$577$	−7.09748e6			−0.887493			−0.443746	+	0.896152i	$0.646351\pi$
$578$		0			0
$579$	−7.07991e6	−	6.06236e6i	−0.877670	−	0.751528i
$580$	3.19484e7			3.94347
$581$		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	$0.5\pi$
$588$	1.51433e6	+	1.29668e6i	0.180625	+	0.154665i
$589$		0			0
$590$		0			0
$591$	−1.08571e7	+	1.26795e7i	−1.27863	+	1.49325i
$592$		0			0
$593$		−	6.57743e6i		−	0.768103i	−0.923312	−	0.384052i	$-0.874528\pi$
$593$		−	6.57743e6i		−	0.768103i	0.923312	−	0.384052i	$-0.125472\pi$
$594$		0			0
$595$	−8.63472e6			−0.999898
$596$		0			0
$597$	1.19119e7	+	1.01999e7i	1.36787	+	1.17128i
$598$		0			0
$599$			7.62940e6i			0.868807i	0.900718	+	0.434403i	$0.143041\pi$
$599$			7.62940e6i			0.868807i	−0.900718	+	0.434403i	$0.856959\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			1.79728e7i			1.99630i
$606$		0			0
$607$	7.27115e6			0.800998			0.400499	−	0.916297i	$-0.368837\pi$
$607$	7.27115e6			0.800998			0.400499	+	0.916297i	$0.368837\pi$
$608$		0			0
$609$	1.02665e7	−	1.19896e7i	1.12170	−	1.30997i
$610$		0			0
$611$		0			0
$612$	−5.25249e6	+	818257.i	−0.566874	+	0.0883102i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	693521.	+	593846.i	0.0739387	+	0.0633120i
$616$		0			0
$617$			2.26641e6i			0.239677i	0.992793	+	0.119839i	$0.0382377\pi$
$617$			2.26641e6i			0.239677i	−0.992793	+	0.119839i	$0.961762\pi$
$618$		0			0
$619$	−5.35363e6			−0.561593			−0.280796	−	0.959767i	$-0.590599\pi$
$619$	−5.35363e6			−0.561593			−0.280796	+	0.959767i	$0.590599\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.81086e7			4.92632
$626$		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
$635$		−	4.05382e7i		−	3.98960i
$636$	−9.89451e6	+	1.15553e7i	−0.969955	+	1.13276i
$637$		0			0
$638$		0			0
$639$	−1.42765e7	+	2.22406e6i	−1.38315	+	0.215474i
$640$		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
$643$	343814.			0.0327941			0.0163971	−	0.999866i	$-0.494780\pi$
$643$	343814.			0.0327941			0.0163971	+	0.999866i	$0.494780\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.15940e7i		−	1.08886i	−0.838805	−	0.544432i	$-0.816745\pi$
$647$		−	1.15940e7i		−	1.08886i	0.838805	−	0.544432i	$-0.183255\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.72026e7			−1.58480
$653$			2.14876e7i			1.97199i	0.166766	+	0.985997i	$0.446668\pi$
$653$			2.14876e7i			1.97199i	−0.166766	+	0.985997i	$0.553332\pi$
$654$		0			0
$655$		0			0
$656$			537441.i			0.0487609i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.84281e7			1.64050			0.820250	−	0.572005i	$-0.193834\pi$
$661$	1.84281e7			1.64050			0.820250	+	0.572005i	$0.193834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			3.86593e7i			3.39000i
$666$		0			0
$667$		0			0
$668$			9.38461e6i			0.813720i
$669$	5.29790e6	+	4.53647e6i	0.457655	+	0.391880i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	1.85783e7	−	3.00597e7i	1.56945	−	2.53936i
$676$	−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	$0.5\pi$
$684$	3.66350e6	+	2.35164e7i	0.299403	+	1.92190i
$685$	−3.41070e7			−2.77726
$686$		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
$695$		−	4.45578e7i		−	3.49914i
$696$		0			0
$697$	358796.			0.0279747
$698$		0			0
$699$		0			0
$700$	3.37876e7			2.60623
$701$		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
$708$	−8.67505e6	+	1.01311e7i	−0.650412	+	0.759581i
$709$	9.67292e6			0.722673			0.361336	−	0.932435i	$-0.382321\pi$
$709$	9.67292e6			0.722673			0.361336	+	0.932435i	$0.382321\pi$
$710$		0			0
$711$	−3.56426e6	−	2.28794e7i	−0.264421	−	1.69735i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	205905.	−	240465.i	0.0149578	−	0.0174685i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	2.74379e7	−	4.27440e6i	1.97251	−	0.307287i
$721$		0			0
$722$		0			0
$723$	−1.23221e7	−	1.05511e7i	−0.876675	−	0.750676i
$724$	−2.05643e7			−1.45804
$725$			8.34592e7i			5.89697i
$726$		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
$735$	−4.52205e6	+	5.28106e6i	−0.308757	+	0.360581i
$736$		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
$749$		−	9.74567e6i		−	0.634756i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−2.95094e6	+	3.44625e6i	−0.189659	+	0.221492i
$754$		0			0
$755$		0			0
$756$	7.21292e6	−	1.16705e7i	0.458994	−	0.742651i
$757$	−3.00805e7			−1.90785			−0.953927	−	0.300039i	$-0.903000\pi$
$757$	−3.00805e7			−1.90785			−0.953927	+	0.300039i	$0.903000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.68229e7i			1.05303i	0.850166	+	0.526514i	$0.176501\pi$
$761$			1.68229e7i			1.05303i	−0.850166	+	0.526514i	$0.823499\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.85359e6	−	1.83175e7i	−0.176294	−	1.13165i
$766$		0			0
$767$		0			0
$768$	1.24159e7	+	1.06314e7i	0.759581	+	0.650412i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	2.11657e7	−	2.47183e7i	1.28232	−	1.49755i
$772$	1.91337e7			1.15546
$773$		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
$779$		−	1.60640e6i		−	0.0948439i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	2.88274e7	+	1.78167e7i	1.68036	+	1.03854i
$784$	−4.09254e6			−0.237795
$785$		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
$789$	2.20913e7	−	2.57992e7i	1.26336	−	1.47542i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−4.02978e7	−	3.45061e7i	−2.26133	−	1.93632i
$796$	−3.21924e7			−1.80082
$797$		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			$\pi$
$812$			3.24025e7i			1.72460i
$813$	2.39397e7	+	2.04991e7i	1.27026	+	1.08770i
$814$		0			0
$815$		−	5.99922e7i		−	3.16374i
$816$	7.09754e6	−	8.28883e6i	0.373149	−	0.435781i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−1.87427e6			−0.0973414
$821$		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
$829$	−3.42994e7			−1.73341			−0.866704	−	0.498823i	$-0.833766\pi$
$829$	−3.42994e7			−1.73341			−0.866704	+	0.498823i	$0.833766\pi$
$830$		0			0
$831$	−2.96109e7	−	2.53551e7i	−1.48747	−	1.27369i
$832$		0			0
$833$			2.73218e6i			0.136426i
$834$		0			0
$835$	−3.27278e7			−1.62443
$836$		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
$841$	−5.95267e7			−2.90216
$842$		0			0
$843$	−2.38562e7	+	2.78604e7i	−1.15620	+	1.35026i
$844$		0			0
$845$		−	4.14351e7i		−	1.99630i
$846$		0			0
$847$	−1.82282e7			−0.873044
$848$		−	3.12286e7i		−	1.49129i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	1.92915e7	−	2.25294e7i	0.910471	−	1.06329i
$853$	4.24393e7			1.99708			0.998541	−	0.0540029i	$-0.0171980\pi$
$853$	4.24393e7			1.99708			0.998541	+	0.0540029i	$0.0171980\pi$
$854$		0			0
$855$	−8.20111e7	+	1.27761e7i	−3.83670	+	0.597698i
$856$		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
$861$	−602288.	+	703379.i	−0.0276883	+	0.0323357i
$862$		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
$867$	1.12785e7	+	9.65751e6i	0.509569	+	0.436332i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			7.83594e7i			3.45996i
$876$		0			0
$877$	4.22854e7			1.85649			0.928243	−	0.371975i	$-0.121319\pi$
$877$	4.22854e7			1.85649			0.928243	+	0.371975i	$0.121319\pi$
$878$		0			0
$879$	3.05780e6	−	3.57104e6i	0.133486	−	0.155892i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	2.29628e7			0.991114			0.495557	−	0.868576i	$-0.334964\pi$
$883$	2.29628e7			0.991114			0.495557	+	0.868576i	$0.334964\pi$
$884$		0			0
$885$	−3.53312e7	−	3.02533e7i	−1.51635	−	1.29842i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	4.11144e7			1.74478
$890$		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
$900$	1.11661e7	+	7.16763e7i	0.459509	+	2.94964i
$901$	−2.08482e7			−0.855572
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7.17160e7i		−	2.91068i
$906$		0			0
$907$	1.90188e7			0.767652			0.383826	−	0.923405i	$-0.374606\pi$
$907$	1.90188e7			0.767652			0.383826	+	0.923405i	$0.374606\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.50133e7i		−	1.39777i	−0.715232	−	0.698887i	$-0.753679\pi$
$911$		−	3.50133e7i		−	1.39777i	0.715232	−	0.698887i	$-0.246321\pi$
$912$	−3.71107e7	−	3.17771e7i	−1.47745	−	1.26511i
$913$		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
$921$	2.13899e7	+	1.83156e7i	0.830919	+	0.711497i
$922$		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
$931$	1.22325e7			0.462531
$932$		0			0
$933$	−3.45605e7	+	4.03614e7i	−1.29980	+	1.51797i
$934$		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
$945$	4.06996e7	+	2.51543e7i	1.48255	+	0.916290i
$946$		0			0
$947$		−	4.64386e7i		−	1.68269i	−0.540499	−	0.841345i	$-0.681764\pi$
$947$		−	4.64386e7i		−	1.68269i	0.540499	−	0.841345i	$-0.318236\pi$
$948$	3.61055e7	+	3.09163e7i	1.30483	+	1.11729i
$949$		0			0
$950$		0			0
$951$	3.19797e7	−	3.73474e7i	1.14663	−	1.33909i
$952$		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
$956$			649868.i			0.0229975i
$957$		0			0
$958$		0			0
$959$		−	3.45918e7i		−	1.21458i
$960$	−3.70760e7	+	4.32991e7i	−1.29842	+	1.51635i
$961$	2.86292e7			1.00000
$962$		0			0
$963$	2.06743e7	−	3.22073e6i	0.718396	−	0.111915i
$964$	3.33010e7			1.15415
$965$			6.67269e7i			2.30666i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−2.12144e7	+	2.47751e7i	−0.725806	+	0.847630i
$970$		0			0
$971$			5.19713e7i			1.76895i	0.466589	+	0.884475i	$0.345483\pi$
$971$			5.19713e7i			1.76895i	−0.466589	+	0.884475i	$0.654517\pi$
$972$	2.71412e7	+	1.14445e7i	0.921434	+	0.388536i
$973$	4.51911e7			1.53028
$974$		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
$980$		−	1.42723e7i		−	0.474710i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	1.19502e8			3.92449
$986$		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
$993$	−1.88985e7	−	1.61823e7i	−0.608210	−	0.520796i
$994$		0			0
$995$		−	1.12268e8i		−	3.59499i
$996$		0			0
$997$	−1.31646e7			−0.419441			−0.209720	−	0.977761i	$-0.567255\pi$
$997$	−1.31646e7			−0.419441			−0.209720	+	0.977761i	$0.567255\pi$
$998$		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.176.6	yes	6
3.2	odd	2	inner	177.6.d.a.176.5	✓	6
59.58	odd	2	CM	177.6.d.a.176.6	yes	6
177.176	even	2	inner	177.6.d.a.176.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.d.a.176.5	✓	6	3.2	odd	2	inner
177.6.d.a.176.5	✓	6	177.176	even	2	inner
177.6.d.a.176.6	yes	6	1.1	even	1	trivial
177.6.d.a.176.6	yes	6	59.58	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +O(q^{10})\)	\(q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +(-378.902 - 324.446i) q^{12} +(1131.47 - 1321.38i) q^{15} +1024.00 q^{16} -683.622i q^{17} -3060.71 q^{19} +3571.09i q^{20} +(1340.17 + 1147.55i) q^{21} -9328.81 q^{25} +(-1991.50 + 3222.24i) q^{27} -3621.85 q^{28} -8946.38i q^{29} -12630.8i q^{35} +(-1196.94 - 7683.33i) q^{36} +524.845i q^{41} +(26794.8 - 4174.22i) q^{45} +(12124.9 + 10382.3i) q^{48} -3996.62 q^{49} +(6931.19 - 8094.56i) q^{51} -30496.7i q^{53} +(-36241.0 - 31032.3i) q^{57} -26738.1i q^{59} +(-36207.0 + 42284.3i) q^{60} +(4233.55 + 27175.7i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(-110460. - 94584.1i) q^{75} +97942.7 q^{76} -95289.8 q^{79} -114275. i q^{80} +(-56250.8 + 17961.9i) q^{81} +(-42885.3 - 36721.7i) q^{84} -76289.9 q^{85} +(90706.7 - 105931. i) q^{87} +341565. i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 192 q^{4}+O(q^{10}) \) 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

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