Embedded newform 177.3.h.a.104.11

• Label: 177.3.h.a.104.11
• Level: $177$
• Weight: $3$
• Character: 177.104
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			104.11
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01300 - 1.68362i) q^{2} +(2.99148 - 0.225943i) q^{3} +(0.0652273 - 0.123032i) q^{4} +(0.831949 + 7.64965i) q^{5} +(-3.41077 - 4.80763i) q^{6} +(5.33496 - 1.79756i) q^{7} +(-8.12120 + 0.440319i) q^{8} +(8.89790 - 1.35181i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01300	−	1.68362i	−0.506500	−	0.841810i	0.493103	−	0.869971i	\(-0.335863\pi\)
							−0.999603	+	0.0281608i	\(0.991035\pi\)
\(3\)	2.99148	−	0.225943i	0.997160	−	0.0753144i
							\(5\)	0.831949	+	7.64965i	0.166390	+	1.52993i	0.718127	+	0.695912i	\(0.245000\pi\)
−0.551737	+	0.834018i	\(0.686035\pi\)
\(7\)	5.33496	−	1.79756i	0.762137	−	0.256794i	0.0887200	−	0.996057i	\(-0.471722\pi\)
							0.673417	+	0.739263i	\(0.264826\pi\)
\(11\)	8.24398	+	5.58956i	0.749453	+	0.508142i	0.875105	−	0.483933i	\(-0.160792\pi\)
							−0.125652	+	0.992074i	\(0.540102\pi\)
\(13\)	5.02967	+	4.76436i	0.386898	+	0.366489i	0.856285	−	0.516503i	\(-0.172767\pi\)
							−0.469388	+	0.882992i	\(0.655525\pi\)
\(17\)	1.16175	−	3.44796i	0.0683384	−	0.202821i	−0.908042	−	0.418879i	\(-0.862423\pi\)
							0.976381	+	0.216057i	\(0.0693199\pi\)
\(19\)	−2.99945	−	18.2958i	−0.157866	−	0.962937i	−0.940335	−	0.340249i	\(-0.889489\pi\)
							0.782470	−	0.622688i	\(-0.213960\pi\)
\(23\)	−28.6827	+	7.96371i	−1.24707	+	0.346248i	−0.827496	−	0.561471i	\(-0.810236\pi\)
							−0.419578	+	0.907720i	\(0.637822\pi\)
\(29\)	21.2538	−	35.3241i	0.732889	−	1.21807i	−0.236282	−	0.971685i	\(-0.575929\pi\)
							0.969171	−	0.246387i	\(-0.0792435\pi\)
\(31\)	2.82788	−	17.2493i	0.0912218	−	0.556428i	−0.901186	−	0.433434i	\(-0.857302\pi\)
							0.992407	−	0.122995i	\(-0.0392498\pi\)
\(37\)	0.226456	−	4.17674i	0.00612044	−	0.112885i	−0.993877	−	0.110495i	\(-0.964756\pi\)
							0.999997	−	0.00239013i	\(-0.000760803\pi\)
\(41\)	−64.7650	−	17.9819i	−1.57963	−	0.438583i	−0.636577	−	0.771213i	\(-0.719650\pi\)
							−0.943057	+	0.332630i	\(0.892064\pi\)
\(43\)	−13.1914	−	19.4559i	−0.306778	−	0.452463i	0.642782	−	0.766049i	\(-0.277780\pi\)
							−0.949560	+	0.313586i	\(0.898470\pi\)
\(47\)	−8.49062	+	78.0699i	−0.180651	+	1.66106i	0.456052	+	0.889953i	\(0.349263\pi\)
							−0.636703	+	0.771109i	\(0.719702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.11	yes	1064
3.2	odd	2	inner	177.3.h.a.104.28	yes	1064
59.21	even	29	inner	177.3.h.a.80.28	yes	1064
177.80	odd	58	inner	177.3.h.a.80.11	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.11	✓	1064	177.80	odd	58	inner
177.3.h.a.80.28	yes	1064	59.21	even	29	inner
177.3.h.a.104.11	yes	1064	1.1	even	1	trivial
177.3.h.a.104.28	yes	1064	3.2	odd	2	inner