Embedded newform 177.3.h.a.107.19

• Label: 177.3.h.a.107.19
• Level: $177$
• Weight: $3$
• Character: 177.107
• 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			107.19
Character	\(\chi\)	\(=\)	177.107
Dual form			177.3.h.a.134.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0841936 - 0.382496i) q^{2} +(-0.169992 + 2.99518i) q^{3} +(3.49109 - 1.61515i) q^{4} +(8.67562 + 2.40877i) q^{5} +(1.15996 - 0.187154i) q^{6} +(-0.763332 + 0.723067i) q^{7} +(-1.85979 - 2.44651i) q^{8} +(-8.94221 - 1.01831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{27}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0841936	−	0.382496i	−0.0420968	−	0.191248i	0.950879	−	0.309561i	\(-0.100182\pi\)
							−0.992976	+	0.118314i	\(0.962251\pi\)
\(3\)	−0.169992	+	2.99518i	−0.0566641	+	0.998393i
							\(5\)	8.67562	+	2.40877i	1.73512	+	0.481755i	0.983964	−	0.178365i	\(-0.0570809\pi\)
0.751160	+	0.660120i	\(0.229495\pi\)
\(7\)	−0.763332	+	0.723067i	−0.109047	+	0.103295i	−0.740281	−	0.672298i	\(-0.765308\pi\)
							0.631234	+	0.775593i	\(0.282549\pi\)
\(11\)	11.0283	−	9.36751i	1.00257	−	0.851592i	0.0136313	−	0.999907i	\(-0.495661\pi\)
							0.988940	+	0.148315i	\(0.0473850\pi\)
\(13\)	−18.4638	+	6.22117i	−1.42029	+	0.478551i	−0.921570	−	0.388212i	\(-0.873093\pi\)
							−0.498720	+	0.866763i	\(0.666196\pi\)
\(17\)	−2.46834	+	2.60579i	−0.145196	+	0.153282i	−0.794537	−	0.607216i	\(-0.792286\pi\)
							0.649341	+	0.760498i	\(0.275045\pi\)
\(19\)	−3.32761	−	8.35168i	−0.175137	−	0.439562i	0.815035	−	0.579411i	\(-0.196717\pi\)
							−0.990173	+	0.139849i	\(0.955338\pi\)
\(23\)	2.96134	+	27.2290i	0.128754	+	1.18387i	0.863077	+	0.505072i	\(0.168534\pi\)
							−0.734324	+	0.678800i	\(0.762501\pi\)
\(29\)	3.59828	−	16.3472i	0.124079	−	0.563695i	−0.873027	−	0.487671i	\(-0.837846\pi\)
							0.997106	−	0.0760238i	\(-0.0242225\pi\)
\(31\)	−15.5432	+	39.0104i	−0.501392	+	1.25840i	0.432755	+	0.901511i	\(0.357541\pi\)
							−0.934147	+	0.356888i	\(0.883838\pi\)
\(37\)	−33.0593	−	25.1310i	−0.893493	−	0.679216i	0.0541547	−	0.998533i	\(-0.482754\pi\)
							−0.947648	+	0.319316i	\(0.896547\pi\)
\(41\)	7.51455	−	69.0951i	0.183282	−	1.68525i	−0.435896	−	0.899997i	\(-0.643569\pi\)
							0.619178	−	0.785251i	\(-0.287466\pi\)
\(43\)	1.60380	−	1.88813i	0.0372976	−	0.0439101i	−0.743197	−	0.669073i	\(-0.766691\pi\)
							0.780495	+	0.625163i	\(0.214967\pi\)
\(47\)	−40.9323	+	11.3648i	−0.870901	+	0.241804i	−0.674101	−	0.738639i	\(-0.735469\pi\)
							−0.196800	+	0.980444i	\(0.563055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.107.19	✓	1064
3.2	odd	2	inner	177.3.h.a.107.20	yes	1064
59.16	even	29	inner	177.3.h.a.134.20	yes	1064
177.134	odd	58	inner	177.3.h.a.134.19	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.107.19	✓	1064	1.1	even	1	trivial
177.3.h.a.107.20	yes	1064	3.2	odd	2	inner
177.3.h.a.134.19	yes	1064	177.134	odd	58	inner
177.3.h.a.134.20	yes	1064	59.16	even	29	inner