Embedded newform 177.2.e.b.19.1

• Label: 177.2.e.b.19.1
• Level: $177$
• Weight: $2$
• Character: 177.19
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			19.1
Character	\(\chi\)	\(=\)	177.19
Dual form			177.2.e.b.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07530 + 2.02823i) q^{2} +(0.725995 - 0.687699i) q^{3} +(-1.83506 - 2.70652i) q^{4} +(1.52678 - 0.336069i) q^{5} +(0.614149 + 2.21197i) q^{6} +(3.14104 - 2.38776i) q^{7} +(2.89830 - 0.315209i) q^{8} +(0.0541389 - 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07530	+	2.02823i	−0.760350	+	1.43417i	0.135946	+	0.990716i	\(0.456593\pi\)
							−0.896296	+	0.443457i	\(0.853752\pi\)
\(3\)	0.725995	−	0.687699i	0.419154	−	0.397043i
							\(5\)	1.52678	−	0.336069i	0.682796	−	0.150295i	0.139990	−	0.990153i	\(-0.455293\pi\)
0.542806	+	0.839858i	\(0.317362\pi\)
\(7\)	3.14104	−	2.38776i	1.18720	−	0.902487i	0.190345	−	0.981717i	\(-0.439039\pi\)
							0.996856	+	0.0792298i	\(0.0252461\pi\)
\(11\)	0.922787	+	2.31602i	0.278231	+	0.698306i	0.999981	+	0.00615283i	\(0.00195852\pi\)
							−0.721750	+	0.692153i	\(0.756662\pi\)
\(13\)	−0.0357213	−	0.658840i	−0.00990730	−	0.182729i	−0.999298	−	0.0374529i	\(-0.988076\pi\)
							0.989391	−	0.145277i	\(-0.0464072\pi\)
\(17\)	2.57713	+	1.95909i	0.625047	+	0.475148i	0.869115	−	0.494610i	\(-0.164689\pi\)
							−0.244068	+	0.969758i	\(0.578482\pi\)
\(19\)	−5.81826	+	1.96040i	−1.33480	+	0.449747i	−0.894084	−	0.447900i	\(-0.852172\pi\)
							−0.440717	+	0.897646i	\(0.645276\pi\)
\(23\)	−2.95769	+	1.77958i	−0.616721	+	0.371069i	−0.789373	−	0.613914i	\(-0.789594\pi\)
							0.172652	+	0.984983i	\(0.444767\pi\)
\(29\)	−2.73526	−	5.15926i	−0.507926	−	0.958050i	−0.996343	−	0.0854397i	\(-0.972771\pi\)
							0.488418	−	0.872610i	\(-0.337574\pi\)
\(31\)	10.0289	+	3.37912i	1.80124	+	0.606908i	0.999963	+	0.00861061i	\(0.00274088\pi\)
							0.801274	+	0.598297i	\(0.204156\pi\)
\(37\)	−0.677494	−	0.0736819i	−0.111379	−	0.0121132i	0.0522596	−	0.998634i	\(-0.483358\pi\)
							−0.163639	+	0.986520i	\(0.552323\pi\)
\(41\)	1.11821	+	0.672805i	0.174635	+	0.105074i	0.600188	−	0.799859i	\(-0.295093\pi\)
							−0.425552	+	0.904934i	\(0.639920\pi\)
\(43\)	0.989690	−	2.48393i	0.150926	−	0.378796i	−0.834002	−	0.551762i	\(-0.813956\pi\)
							0.984928	+	0.172966i	\(0.0553349\pi\)
\(47\)	−9.69483	−	2.13399i	−1.41414	−	0.311275i	−0.558780	−	0.829316i	\(-0.688730\pi\)
							−0.855356	+	0.518041i	\(0.826661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.19.1	✓	140
3.2	odd	2		531.2.i.b.19.5		140
59.28	even	29	inner	177.2.e.b.28.1	yes	140
177.146	odd	58		531.2.i.b.28.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.19.1	✓	140	1.1	even	1	trivial
177.2.e.b.28.1	yes	140	59.28	even	29	inner
531.2.i.b.19.5		140	3.2	odd	2
531.2.i.b.28.5		140	177.146	odd	58