LMFDB - Embedded newform 177.6.d.a.176.1

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			176.1
Root			$-0.422380 - 1.67976i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.6.d.a.176.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +O(q^{10})$	$q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +(470.429 + 165.916i) q^{12} +(-319.824 + 906.809i) q^{15} +1024.00 q^{16} +683.622i q^{17} +2164.66 q^{19} +1973.88i q^{20} +(-2137.97 - 754.045i) q^{21} -679.900 q^{25} +(-1991.50 - 3222.24i) q^{27} -4653.80 q^{28} -5700.02i q^{29} -8970.77i q^{35} +(-6055.48 - 4878.25i) q^{36} -18375.2i q^{41} +(9403.41 - 11672.7i) q^{45} +(-15053.7 - 5309.33i) q^{48} +4343.27 q^{49} +(3544.50 - 10049.9i) q^{51} +8353.58i q^{53} +(-31822.4 - 11223.5i) q^{57} +26738.1i q^{59} +(10234.4 - 29017.9i) q^{60} +(27520.5 + 22170.3i) q^{63} -32768.0 q^{64} -21875.9i q^{68} -59459.7i q^{71} +(9995.15 + 3525.21i) q^{75} -69269.0 q^{76} -1559.45 q^{79} -63164.3i q^{80} +(12569.9 + 57695.6i) q^{81} +(68415.2 + 24129.4i) q^{84} +42168.5 q^{85} +(-29553.9 + 83795.5i) q^{87} -133524. i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.5\pi$
$3$	−14.7009	−	5.18489i	−0.943064	−	0.332611i
$3$	−14.7009	−	5.18489i	−0.943064	−	0.332611i
$4$	−32.0000			−1.00000
$5$		−	61.6839i		−	1.10343i	−0.834031	−	0.551717i	$-0.813973\pi$
$5$		−	61.6839i		−	1.10343i	0.834031	−	0.551717i	$-0.186027\pi$
$6$		0			0
$7$	145.431			1.12179			0.560897	−	0.827886i	$-0.310456\pi$
$7$	145.431			1.12179			0.560897	+	0.827886i	$0.310456\pi$
$8$		0			0
$9$	189.234	+	152.445i	0.778740	+	0.627347i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	470.429	+	165.916i	0.943064	+	0.332611i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−319.824	+	906.809i	−0.367014	+	1.04061i
$16$	1024.00			1.00000
$17$			683.622i			0.573712i	0.957974	+	0.286856i	$0.0926101\pi$
$17$			683.622i			0.573712i	−0.957974	+	0.286856i	$0.907390\pi$
$18$		0			0
$19$	2164.66			1.37564			0.687821	−	0.725881i	$-0.258568\pi$
$19$	2164.66			1.37564			0.687821	+	0.725881i	$0.258568\pi$
$20$			1973.88i			1.10343i
$21$	−2137.97	−	754.045i	−1.05792	−	0.373120i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−679.900			−0.217568
$26$		0			0
$27$	−1991.50	−	3222.24i	−0.525740	−	0.850645i
$28$	−4653.80			−1.12179
$29$		−	5700.02i		−	1.25858i	−0.777170	−	0.629290i	$-0.783346\pi$
$29$		−	5700.02i		−	1.25858i	0.777170	−	0.629290i	$-0.216654\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	8970.77i		−	1.23783i
$36$	−6055.48	−	4878.25i	−0.778740	−	0.627347i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	18375.2i		−	1.70716i	−0.520965	−	0.853578i	$-0.674428\pi$
$41$		−	18375.2i		−	1.70716i	0.520965	−	0.853578i	$-0.325572\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	9403.41	−	11672.7i	0.692236	−	0.859289i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−15053.7	−	5309.33i	−0.943064	−	0.332611i
$49$	4343.27			0.258420
$50$		0			0
$51$	3544.50	−	10049.9i	0.190823	−	0.541047i
$52$		0			0
$53$			8353.58i			0.408492i	0.978920	+	0.204246i	$0.0654742\pi$
$53$			8353.58i			0.408492i	−0.978920	+	0.204246i	$0.934526\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−31822.4	−	11223.5i	−1.29732	−	0.457553i
$58$		0			0
$59$			26738.1i			1.00000i
$59$			26738.1i			1.00000i
$60$	10234.4	−	29017.9i	0.367014	−	1.04061i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	27520.5	+	22170.3i	0.873586	+	0.703753i
$64$	−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	$-0.753220\pi$
$71$		−	59459.7i		−	1.39984i	0.714223	−	0.699918i	$-0.246780\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	9995.15	+	3525.21i	0.205181	+	0.0723655i
$76$	−69269.0			−1.37564
$77$		0			0
$78$		0			0
$79$	−1559.45			−0.0281127			−0.0140564	−	0.999901i	$-0.504474\pi$
$79$	−1559.45			−0.0281127			−0.0140564	+	0.999901i	$0.504474\pi$
$80$		−	63164.3i		−	1.10343i
$81$	12569.9	+	57695.6i	0.212873	+	0.977080i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	68415.2	+	24129.4i	1.05792	+	0.373120i
$85$	42168.5			0.633053
$86$		0			0
$87$	−29553.9	+	83795.5i	−0.418617	+	1.18692i
$88$		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
$95$		−	133524.i		−	1.51793i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	21756.8			0.217568
$101$		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
$105$	−46512.4	+	131878.i	−0.411714	+	1.16735i
$106$		0			0
$107$		−	234144.i		−	1.97708i	−0.150964	−	0.988539i	$-0.548238\pi$
$107$		−	234144.i		−	1.97708i	0.150964	−	0.988539i	$-0.451762\pi$
$108$	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
$112$	148922.			1.12179
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			182401.i			1.25858i
$117$		0			0
$118$		0			0
$119$			99420.0i			0.643586i
$120$		0			0
$121$	−161051.			−1.00000
$122$		0			0
$123$	−95273.5	+	270133.i	−0.567818	+	1.60996i
$124$		0			0
$125$		−	150823.i		−	0.863363i
$126$		0			0
$127$	−193827.			−1.06636			−0.533182	−	0.846001i	$-0.679004\pi$
$127$	−193827.			−1.06636			−0.533182	+	0.846001i	$0.679004\pi$
$128$		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
$133$	314809.			1.54319
$134$		0			0
$135$	−198760.	+	122843.i	−0.938632	+	0.580120i
$136$		0			0
$137$			120553.i			0.548750i	0.961623	+	0.274375i	$0.0884710\pi$
$137$			120553.i			0.548750i	−0.961623	+	0.274375i	$0.911529\pi$
$138$		0			0
$139$	399275.			1.75281			0.876406	−	0.481574i	$-0.159935\pi$
$139$	399275.			1.75281			0.876406	+	0.481574i	$0.159935\pi$
$140$			287065.i			1.23783i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	193775.	+	156104.i	0.778740	+	0.627347i
$145$	−351599.			−1.38876
$146$		0			0
$147$	−63850.1	−	22519.4i	−0.243707	−	0.0859534i
$148$		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
$153$	−104215.	+	129364.i	−0.359916	+	0.446772i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	43312.4	−	122805.i	0.135869	−	0.385234i
$160$		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	$0.208831\pi$
$164$			588007.i			1.70716i
$165$		0			0
$166$		0			0
$167$		−	716872.i		−	1.98907i	−0.104394	−	0.994536i	$-0.533290\pi$
$167$		−	716872.i		−	1.98907i	0.104394	−	0.994536i	$-0.466710\pi$
$168$		0			0
$169$	371293.			1.00000
$170$		0			0
$171$	409627.	+	329992.i	1.07127	+	0.863004i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−98878.8			−0.244066
$176$		0			0
$177$	138634.	−	393074.i	0.332611	−	0.943064i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−300909.	+	373526.i	−0.692236	+	0.859289i
$181$	201231.			0.456561			0.228281	−	0.973595i	$-0.426690\pi$
$181$	201231.			0.456561			0.228281	+	0.973595i	$0.426690\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−289626.	−	468615.i	−0.589771	−	0.954248i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	481720.	+	169898.i	0.943064	+	0.332611i
$193$	−432620.			−0.836013			−0.418007	−	0.908444i	$-0.637271\pi$
$193$	−432620.			−0.836013			−0.418007	+	0.908444i	$0.637271\pi$
$194$		0			0
$195$		0			0
$196$	−138985.			−0.258420
$197$		−	1.07084e6i		−	1.96588i	−0.183921	−	0.982941i	$-0.558879\pi$
$197$		−	1.07084e6i		−	1.96588i	0.183921	−	0.982941i	$-0.441121\pi$
$198$		0			0
$199$	−923948.			−1.65392			−0.826961	−	0.562260i	$-0.809932\pi$
$199$	−923948.			−1.65392			−0.826961	+	0.562260i	$0.809932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	828961.i		−	1.41187i
$204$	−113424.	+	321596.i	−0.190823	+	0.541047i
$205$	−1.13346e6			−1.88373
$206$		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			$\pi$
$212$		−	267315.i		−	0.408492i
$213$	−308292.	+	874113.i	−0.465601	+	1.32014i
$214$		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
$225$	−128660.	−	103648.i	−0.169429	−	0.136491i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	1.01832e6	+	359152.i	1.29732	+	0.457553i
$229$		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
$237$	22925.3	+	8085.56i	0.0265121	+	0.00935060i
$238$		0			0
$239$		−	1.53957e6i		−	1.74344i	−0.490009	−	0.871718i	$-0.663006\pi$
$239$		−	1.53957e6i		−	1.74344i	0.490009	−	0.871718i	$-0.336994\pi$
$240$	−327500.	+	928573.i	−0.367014	+	1.04061i
$241$	−755113.			−0.837470			−0.418735	−	0.908108i	$-0.637526\pi$
$241$	−755113.			−0.837470			−0.418735	+	0.908108i	$0.637526\pi$
$242$		0			0
$243$	114356.	−	913352.i	0.124235	−	0.992253i
$244$		0			0
$245$		−	267910.i		−	0.285150i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.56480e6i		−	1.56774i	−0.620923	−	0.783872i	$-0.713242\pi$
$251$		−	1.56480e6i		−	1.56774i	0.620923	−	0.783872i	$-0.286758\pi$
$252$	−880657.	−	709450.i	−0.873586	−	0.703753i
$253$		0			0
$254$		0			0
$255$	−619915.	−	218639.i	−0.597010	−	0.210560i
$256$	1.04858e6			1.00000
$257$		−	735549.i		−	0.694671i	−0.937741	−	0.347335i	$-0.887087\pi$
$257$		−	735549.i		−	0.694671i	0.937741	−	0.347335i	$-0.112913\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	868940.	−	1.07864e6i	0.789566	−	0.980107i
$262$		0			0
$263$		−	626525.i		−	0.558533i	−0.960214	−	0.279267i	$-0.909909\pi$
$263$		−	626525.i		−	0.558533i	0.960214	−	0.279267i	$-0.0900913\pi$
$264$		0			0
$265$	515281.			0.450744
$266$		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
$271$	−2.15946e6			−1.78617			−0.893083	−	0.449891i	$-0.851463\pi$
$271$	−2.15946e6			−1.78617			−0.893083	+	0.449891i	$0.851463\pi$
$272$			700029.i			0.573712i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.69981e6			1.33107			0.665536	−	0.746365i	$-0.268203\pi$
$277$	1.69981e6			1.33107			0.665536	+	0.746365i	$0.268203\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.22706e6i			1.68254i	0.540614	+	0.841271i	$0.318192\pi$
$281$			2.22706e6i			1.68254i	−0.540614	+	0.841271i	$0.681808\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			1.90271e6i			1.39984i
$285$	−692309.	+	1.96293e6i	−0.504880	+	1.43151i
$286$		0			0
$287$		−	2.67233e6i		−	1.91508i
$288$		0			0
$289$	952518.			0.670855
$290$		0			0
$291$		0			0
$292$		0			0
$293$			2.38101e6i			1.62029i	0.586229	+	0.810145i	$0.300612\pi$
$293$			2.38101e6i			1.62029i	−0.586229	+	0.810145i	$0.699388\pi$
$294$		0			0
$295$	1.64931e6			1.10343
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−319845.	−	112807.i	−0.205181	−	0.0723655i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2.21661e6			1.37564
$305$		0			0
$306$		0			0
$307$	−3.29773e6			−1.99696			−0.998481	−	0.0551001i	$-0.982452\pi$
$307$	−3.29773e6			−1.99696			−0.998481	+	0.0551001i	$0.982452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.82160e6i			1.06796i	0.845499	+	0.533978i	$0.179303\pi$
$311$			1.82160e6i			1.06796i	−0.845499	+	0.533978i	$0.820697\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	1.36755e6	−	1.69757e6i	0.776546	−	0.963945i
$316$	49902.3			0.0281127
$317$			3.15415e6i			1.76293i	0.472250	+	0.881465i	$0.343442\pi$
$317$			3.15415e6i			1.76293i	−0.472250	+	0.881465i	$0.656558\pi$
$318$		0			0
$319$		0			0
$320$			2.02126e6i			1.10343i
$321$	−1.21401e6	+	3.44213e6i	−0.657597	+	1.86451i
$322$		0			0
$323$			1.47981e6i			0.789221i
$324$	−402237.	−	1.84626e6i	−0.212873	−	0.977080i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.96174e6			1.98754			0.993769	−	0.111459i	$-0.0355523\pi$
$331$	3.96174e6			1.98754			0.993769	+	0.111459i	$0.0355523\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−2.18929e6	−	772142.i	−1.05792	−	0.373120i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	−1.34939e6			−0.633053
$341$		0			0
$342$		0			0
$343$	−1.81262e6			−0.831899
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	945726.	−	2.68145e6i	0.418617	−	1.18692i
$349$		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
$355$	−3.66771e6			−1.54463
$356$		0			0
$357$	515482.	−	1.46157e6i	0.214064	−	0.606943i
$358$		0			0
$359$		−	4.22397e6i		−	1.72976i	−0.501981	−	0.864878i	$-0.667395\pi$
$359$		−	4.22397e6i		−	1.72976i	0.501981	−	0.864878i	$-0.332605\pi$
$360$		0			0
$361$	2.20964e6			0.892389
$362$		0			0
$363$	2.36760e6	+	835031.i	0.943064	+	0.332611i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	2.80122e6	−	3.47722e6i	1.07098	−	1.32943i
$370$		0			0
$371$			1.21487e6i			0.458243i
$372$		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
$375$	−782002.	+	2.21724e6i	−0.287164	+	0.814206i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−5.44338e6			−1.94657			−0.973287	−	0.229591i	$-0.926261\pi$
$379$	−5.44338e6			−1.94657			−0.973287	+	0.229591i	$0.926261\pi$
$380$			4.27278e6i			1.51793i
$381$	2.84944e6	+	1.00497e6i	1.00565	+	0.354684i
$382$		0			0
$383$			5.18018e6i			1.80446i	0.431252	+	0.902231i	$0.358072\pi$
$383$			5.18018e6i			1.80446i	−0.431252	+	0.902231i	$0.641928\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			2.43319e6i			0.815273i	0.913144	+	0.407636i	$0.133647\pi$
$389$			2.43319e6i			0.815273i	−0.913144	+	0.407636i	$0.866353\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			96192.8i			0.0310206i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−4.62798e6	−	1.63225e6i	−1.45532	−	0.513280i
$400$	−696218.			−0.217568
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3.55889e6	−	775361.i	1.07814	−	0.234891i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	625051.	−	1.77223e6i	0.182520	−	0.517507i
$412$		0			0
$413$			3.88855e6i			1.12179i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.86971e6	−	2.07020e6i	−1.65301	−	0.583004i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	1.48840e6	−	4.22011e6i	0.411714	−	1.16735i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	464795.i		−	0.124821i
$426$		0			0
$427$		0			0
$428$			7.49261e6i			1.97708i
$429$		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
$433$	7.80260e6			1.99995			0.999976	−	0.00698713i	$-0.00222409\pi$
$433$	7.80260e6			1.99995			0.999976	+	0.00698713i	$0.00222409\pi$
$434$		0			0
$435$	5.16883e6	+	1.82300e6i	1.30969	+	0.461917i
$436$		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
$441$	821894.	+	662111.i	0.201242	+	0.162119i
$442$		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
$448$	−4.76549e6			−1.12179
$449$		−	1.57248e6i		−	0.368102i	−0.982917	−	0.184051i	$-0.941079\pi$
$449$		−	1.57248e6i		−	0.368102i	0.982917	−	0.184051i	$-0.0589212\pi$
$450$		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	$0.268990\pi$
$461$			6.82636e6i			1.49602i	−0.663689	+	0.748009i	$0.731010\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		−	5.83682e6i		−	1.25858i
$465$		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
$475$	−1.47175e6			−0.299296
$476$		−	3.18144e6i		−	0.643586i
$477$	−1.27346e6	+	1.58078e6i	−0.256266	+	0.318109i
$478$		0			0
$479$		−	8.05733e6i		−	1.60455i	−0.596957	−	0.802273i	$-0.703624\pi$
$479$		−	8.05733e6i		−	1.60455i	0.596957	−	0.802273i	$-0.296376\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	5.15363e6			1.00000
$485$		0			0
$486$		0			0
$487$	1.64860e6			0.314988			0.157494	−	0.987520i	$-0.449658\pi$
$487$	1.64860e6			0.314988			0.157494	+	0.987520i	$0.449658\pi$
$488$		0			0
$489$	−7.90293e6	−	2.78730e6i	−1.49457	−	0.527122i
$490$		0			0
$491$		−	1.03035e7i		−	1.92877i	−0.264499	−	0.964386i	$-0.585207\pi$
$491$		−	1.03035e7i		−	1.92877i	0.264499	−	0.964386i	$-0.414793\pi$
$492$	3.04875e6	−	8.64425e6i	0.567818	−	1.60996i
$493$	3.89666e6			0.722063
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.64731e6i		−	1.57033i
$498$		0			0
$499$	7.45662e6			1.34057			0.670286	−	0.742102i	$-0.266171\pi$
$499$	7.45662e6			1.34057			0.670286	+	0.742102i	$0.266171\pi$
$500$			4.82634e6i			0.863363i
$501$	−3.71690e6	+	1.05387e7i	−0.661587	+	1.87582i
$502$		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
$507$	−5.45835e6	−	1.92511e6i	−0.943064	−	0.332611i
$508$	6.20247e6			1.06636
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4.31092e6	−	6.97505e6i	−0.723229	−	1.17018i
$514$		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	$-0.730928\pi$
$521$		−	9.27107e6i		−	1.49636i	0.663497	−	0.748179i	$-0.269072\pi$
$522$		0			0
$523$	−7.19380e6			−1.15002			−0.575009	−	0.818147i	$-0.695001\pi$
$523$	−7.19380e6			−1.15002			−0.575009	+	0.818147i	$0.695001\pi$
$524$		0			0
$525$	1.45361e6	+	512675.i	0.230170	+	0.0811791i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.43634e6			−1.00000
$530$		0			0
$531$	−4.07609e6	+	5.05975e6i	−0.627347	+	0.778740i
$532$	−1.00739e7			−1.54319
$533$		0			0
$534$		0			0
$535$	−1.44429e7			−2.18158
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	6.36033e6	−	3.93099e6i	0.938632	−	0.580120i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−2.95828e6	−	1.04336e6i	−0.430566	−	0.151857i
$544$		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	$0.226456\pi$
$548$		−	3.85768e6i		−	0.548750i
$549$		0			0
$550$		0			0
$551$		−	1.23386e7i		−	1.73136i
$552$		0			0
$553$	−226793.			−0.0315367
$554$		0			0
$555$		0			0
$556$	−1.27768e7			−1.75281
$557$			7.22645e6i			0.986931i	0.869765	+	0.493466i	$0.164270\pi$
$557$			7.22645e6i			0.986931i	−0.869765	+	0.493466i	$0.835730\pi$
$558$		0			0
$559$		0			0
$560$		−	9.18607e6i		−	1.23783i
$561$		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
$567$	1.82806e6	+	8.39075e6i	0.238799	+	1.09608i
$568$		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
$576$	−6.20082e6	−	4.99532e6i	−0.778740	−	0.627347i
$577$	1.59619e7			1.99593			0.997964	−	0.0637806i	$-0.0203158\pi$
$577$	1.59619e7			1.99593			0.997964	+	0.0637806i	$0.0203158\pi$
$578$		0			0
$579$	6.35991e6	+	2.24308e6i	0.788414	+	0.278067i
$580$	1.12512e7			1.38876
$581$		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	$0.5\pi$
$588$	2.04320e6	+	720620.i	0.243707	+	0.0859534i
$589$		0			0
$590$		0			0
$591$	−5.55217e6	+	1.57423e7i	−0.653873	+	1.85395i
$592$		0			0
$593$			1.04058e7i			1.21517i	0.794254	+	0.607586i	$0.207862\pi$
$593$			1.04058e7i			1.21517i	−0.794254	+	0.607586i	$0.792138\pi$
$594$		0			0
$595$	6.13261e6			0.710155
$596$		0			0
$597$	1.35829e7	+	4.79057e6i	1.55975	+	0.550112i
$598$		0			0
$599$			1.75146e7i			1.99449i	0.0741549	+	0.997247i	$0.476374\pi$
$599$			1.75146e7i			1.99449i	−0.0741549	+	0.997247i	$0.523626\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			9.93425e6i			1.10343i
$606$		0			0
$607$	−1.80424e7			−1.98757			−0.993786	−	0.111307i	$-0.964496\pi$
$607$	−1.80424e7			−1.98757			−0.993786	+	0.111307i	$0.964496\pi$
$608$		0			0
$609$	−4.29807e6	+	1.21865e7i	−0.469602	+	1.33148i
$610$		0			0
$611$		0			0
$612$	3.33488e6	−	4.13966e6i	0.359916	−	0.446772i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	1.66628e7	+	5.87684e6i	1.77648	+	0.626550i
$616$		0			0
$617$			1.73937e7i			1.83941i	0.392613	+	0.919704i	$0.371571\pi$
$617$			1.73937e7i			1.83941i	−0.392613	+	0.919704i	$0.628429\pi$
$618$		0			0
$619$	1.85240e7			1.94316			0.971581	−	0.236707i	$-0.0760681\pi$
$619$	1.85240e7			1.94316			0.971581	+	0.236707i	$0.0760681\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.14280e7			−1.17023
$626$		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
$635$			1.19560e7i			1.17666i
$636$	−1.38600e6	+	3.92977e6i	−0.135869	+	0.385234i
$637$		0			0
$638$		0			0
$639$	9.06435e6	−	1.12518e7i	0.878182	−	1.09011i
$640$		0			0
$641$		−	1.91009e7i		−	1.83615i	−0.396404	−	0.918076i	$-0.629742\pi$
$641$		−	1.91009e7i		−	1.83615i	0.396404	−	0.918076i	$-0.370258\pi$
$642$		0			0
$643$	1.79845e7			1.71542			0.857710	−	0.514133i	$-0.171886\pi$
$643$	1.79845e7			1.71542			0.857710	+	0.514133i	$0.171886\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.67270e6i			0.908420i	0.890895	+	0.454210i	$0.150078\pi$
$647$			9.67270e6i			0.908420i	−0.890895	+	0.454210i	$0.849922\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.72026e7			−1.58480
$653$			1.38912e7i			1.27484i	0.770515	+	0.637422i	$0.219999\pi$
$653$			1.38912e7i			1.27484i	−0.770515	+	0.637422i	$0.780001\pi$
$654$		0			0
$655$		0			0
$656$		−	1.88162e7i		−	1.70716i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−2.03432e7			−1.81099			−0.905496	−	0.424355i	$-0.860501\pi$
$661$	−2.03432e7			−1.81099			−0.905496	+	0.424355i	$0.860501\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.94186e7i		−	1.70280i
$666$		0			0
$667$		0			0
$668$			2.29399e7i			1.98907i
$669$	−6.57765e6	−	2.31988e6i	−0.568205	−	0.200401i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	1.35402e6	+	2.19080e6i	0.114384	+	0.185073i
$676$	−1.18814e7			−1.00000
$677$			388942.i			0.0326147i	0.999867	+	0.0163074i	$0.00519102\pi$
$677$			388942.i			0.0326147i	−0.999867	+	0.0163074i	$0.994809\pi$
$678$		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	$0.5\pi$
$684$	−1.31080e7	−	1.05597e7i	−1.07127	−	0.863004i
$685$	7.43615e6			0.605510
$686$		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
$695$		−	2.46288e7i		−	1.93411i
$696$		0			0
$697$	1.25617e7			0.979415
$698$		0			0
$699$		0			0
$700$	3.16412e6			0.244066
$701$		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
$708$	−4.43628e6	+	1.25784e7i	−0.332611	+	0.943064i
$709$	−2.64534e7			−1.97636			−0.988181	−	0.153291i	$-0.951013\pi$
$709$	−2.64534e7			−1.97636			−0.988181	+	0.153291i	$0.951013\pi$
$710$		0			0
$711$	−295100.	−	237730.i	−0.0218925	−	0.0176364i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.98252e6	+	2.26331e7i	−0.579885	+	1.64417i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	9.62909e6	−	1.19528e7i	0.692236	−	0.859289i
$721$		0			0
$722$		0			0
$723$	1.11009e7	+	3.91518e6i	0.789788	+	0.278552i
$724$	−6.43940e6			−0.456561
$725$			3.87544e6i			0.273827i
$726$		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
$735$	−1.38908e6	+	3.93852e6i	−0.0948439	+	0.268915i
$736$		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	$-0.763956\pi$
$743$		−	2.03276e7i		−	1.35087i	0.737419	−	0.675436i	$-0.236044\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	3.40519e7i		−	2.21787i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−8.11332e6	+	2.30040e7i	−0.521448	+	1.47848i
$754$		0			0
$755$		0			0
$756$	9.26805e6	+	1.49957e7i	0.589771	+	0.954248i
$757$	2.32339e7			1.47361			0.736805	−	0.676105i	$-0.236334\pi$
$757$	2.32339e7			1.47361			0.736805	+	0.676105i	$0.236334\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.51133e7i		−	0.946017i	−0.881058	−	0.473008i	$-0.843168\pi$
$761$		−	1.51133e7i		−	0.946017i	0.881058	−	0.473008i	$-0.156832\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	7.97970e6	+	6.42838e6i	0.492984	+	0.397144i
$766$		0			0
$767$		0			0
$768$	−1.54150e7	−	5.43675e6i	−0.943064	−	0.332611i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−3.81374e6	+	1.08132e7i	−0.231055	+	0.655119i
$772$	1.38438e7			0.836013
$773$		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
$779$		−	3.97761e7i		−	2.34843i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.83668e7	+	1.13516e7i	−1.07061	+	0.661686i
$784$	4.44751e6			0.258420
$785$		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
$789$	−3.24846e6	+	9.21049e6i	−0.185774	+	0.526733i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−7.57511e6	−	2.67168e6i	−0.425080	−	0.149922i
$796$	2.95663e7			1.65392
$797$		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			$\pi$
$812$			2.65267e7i			1.41187i
$813$	3.17460e7	+	1.11966e7i	1.68447	+	0.594098i
$814$		0			0
$815$		−	3.31601e7i		−	1.74873i
$816$	3.62957e6	−	1.02911e7i	0.190823	−	0.541047i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	3.62706e7			1.88373
$821$		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	$-0.786812\pi$
$827$		−	2.44196e7i		−	1.24158i	0.783977	−	0.620790i	$-0.213188\pi$
$828$		0			0
$829$	53752.1			0.00271650			0.00135825	−	0.999999i	$-0.499568\pi$
$829$	53752.1			0.00271650			0.00135825	+	0.999999i	$0.499568\pi$
$830$		0			0
$831$	−2.49888e7	−	8.81334e6i	−1.25529	−	0.442729i
$832$		0			0
$833$			2.96916e6i			0.148259i
$834$		0			0
$835$	−4.42194e7			−2.19481
$836$		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
$841$	−1.19790e7			−0.584026
$842$		0			0
$843$	1.15471e7	−	3.27398e7i	0.559631	−	1.58675i
$844$		0			0
$845$		−	2.29028e7i		−	1.10343i
$846$		0			0
$847$	−2.34219e7			−1.12179
$848$			8.55407e6i			0.408492i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	9.86535e6	−	2.79716e7i	0.465601	−	1.32014i
$853$	−1.92320e7			−0.905005			−0.452502	−	0.891763i	$-0.649469\pi$
$853$	−1.92320e7			−0.905005			−0.452502	+	0.891763i	$0.649469\pi$
$854$		0			0
$855$	2.03552e7	−	2.52674e7i	0.952268	−	1.18207i
$856$		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
$861$	−1.38558e7	+	3.92858e7i	−0.636975	+	1.80604i
$862$		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
$867$	−1.40029e7	−	4.93870e6i	−0.632659	−	0.223134i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.19344e7i		−	0.968514i
$876$		0			0
$877$	−6.46788e6			−0.283964			−0.141982	−	0.989869i	$-0.545347\pi$
$877$	−6.46788e6			−0.283964			−0.141982	+	0.989869i	$0.545347\pi$
$878$		0			0
$879$	1.23453e7	−	3.50031e7i	0.538926	−	1.52804i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−4.63368e7			−1.99997			−0.999987	−	0.00512297i	$-0.998369\pi$
$883$	−4.63368e7			−1.99997			−0.999987	+	0.00512297i	$0.998369\pi$
$884$		0			0
$885$	−2.42463e7	−	8.55148e6i	−1.04061	−	0.367014i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−2.81885e7			−1.19624
$890$		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
$900$	4.11712e6	+	3.31672e6i	0.169429	+	0.136491i
$901$	−5.71069e6			−0.234356
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.24127e7i		−	0.503785i
$906$		0			0
$907$	3.01158e7			1.21556			0.607780	−	0.794106i	$-0.292060\pi$
$907$	3.01158e7			1.21556			0.607780	+	0.794106i	$0.292060\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.35249e7i			0.539931i	0.962870	+	0.269966i	$0.0870123\pi$
$911$			1.35249e7i			0.539931i	−0.962870	+	0.269966i	$0.912988\pi$
$912$	−3.25862e7	−	1.14929e7i	−1.29732	−	0.457553i
$913$		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
$921$	4.84797e7	+	1.70984e7i	1.88326	+	0.664211i
$922$		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
$931$	9.40169e6			0.355494
$932$		0			0
$933$	9.44482e6	−	2.67793e7i	0.355213	−	1.00715i
$934$		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
$945$	−2.89060e7	+	1.78653e7i	−1.05295	+	0.650774i
$946$		0			0
$947$		−	4.90556e7i		−	1.77752i	−0.458376	−	0.888758i	$-0.651569\pi$
$947$		−	4.90556e7i		−	1.77752i	0.458376	−	0.888758i	$-0.348431\pi$
$948$	−733610.	−	258738.i	−0.0265121	−	0.00935060i
$949$		0			0
$950$		0			0
$951$	1.63539e7	−	4.63690e7i	0.586369	−	1.66256i
$952$		0			0
$953$			4.95637e7i			1.76779i	0.467681	+	0.883897i	$0.345089\pi$
$953$			4.95637e7i			1.76779i	−0.467681	+	0.883897i	$0.654911\pi$
$954$		0			0
$955$		0			0
$956$			4.92664e7i			1.74344i
$957$		0			0
$958$		0			0
$959$			1.75321e7i			0.615585i
$960$	1.04800e7	−	2.97143e7i	0.367014	−	1.04061i
$961$	2.86292e7			1.00000
$962$		0			0
$963$	3.56942e7	−	4.43080e7i	1.24031	−	1.53963i
$964$	2.41636e7			0.837470
$965$			2.66857e7i			0.922486i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	7.67264e6	−	2.17545e7i	0.262504	−	0.744287i
$970$		0			0
$971$			2.24225e6i			0.0763195i	0.999272	+	0.0381597i	$0.0121496\pi$
$971$			2.24225e6i			0.0763195i	−0.999272	+	0.0381597i	$0.987850\pi$
$972$	−3.65939e6	+	2.92273e7i	−0.124235	+	0.992253i
$973$	5.80671e7			1.96629
$974$		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
$980$			8.57311e6i			0.285150i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−6.60533e7			−2.16922
$986$		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
$993$	−5.82411e7	−	2.05412e7i	−1.87438	−	0.661076i
$994$		0			0
$995$			5.69927e7i			1.82499i
$996$		0			0
$997$	5.97358e7			1.90325			0.951626	−	0.307258i	$-0.0994113\pi$
$997$	5.97358e7			1.90325			0.951626	+	0.307258i	$0.0994113\pi$
$998$		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.176.1	✓	6
3.2	odd	2	inner	177.6.d.a.176.2	yes	6
59.58	odd	2	CM	177.6.d.a.176.1	✓	6
177.176	even	2	inner	177.6.d.a.176.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.d.a.176.1	✓	6	1.1	even	1	trivial
177.6.d.a.176.1	✓	6	59.58	odd	2	CM
177.6.d.a.176.2	yes	6	3.2	odd	2	inner
177.6.d.a.176.2	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)

\(f(q)\)	\(=\)	\(q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +O(q^{10})\)	\(q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +(470.429 + 165.916i) q^{12} +(-319.824 + 906.809i) q^{15} +1024.00 q^{16} +683.622i q^{17} +2164.66 q^{19} +1973.88i q^{20} +(-2137.97 - 754.045i) q^{21} -679.900 q^{25} +(-1991.50 - 3222.24i) q^{27} -4653.80 q^{28} -5700.02i q^{29} -8970.77i q^{35} +(-6055.48 - 4878.25i) q^{36} -18375.2i q^{41} +(9403.41 - 11672.7i) q^{45} +(-15053.7 - 5309.33i) q^{48} +4343.27 q^{49} +(3544.50 - 10049.9i) q^{51} +8353.58i q^{53} +(-31822.4 - 11223.5i) q^{57} +26738.1i q^{59} +(10234.4 - 29017.9i) q^{60} +(27520.5 + 22170.3i) q^{63} -32768.0 q^{64} -21875.9i q^{68} -59459.7i q^{71} +(9995.15 + 3525.21i) q^{75} -69269.0 q^{76} -1559.45 q^{79} -63164.3i q^{80} +(12569.9 + 57695.6i) q^{81} +(68415.2 + 24129.4i) q^{84} +42168.5 q^{85} +(-29553.9 + 83795.5i) q^{87} -133524. i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 192 q^{4}+O(q^{10}) \) 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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