Embedded newform 177.2.e.b.4.1

• Label: 177.2.e.b.4.1
• Level: $177$
• Weight: $2$
• Character: 177.4
• 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			4.1
Character	\(\chi\)	\(=\)	177.4
Dual form			177.2.e.b.133.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.53795 - 0.276019i) q^{2} +(-0.647386 + 0.762162i) q^{3} +(4.41176 + 0.971102i) q^{4} +(-0.0920748 - 0.0699934i) q^{5} +(1.85340 - 1.75564i) q^{6} +(0.00524223 - 0.0131570i) q^{7} +(-6.09023 - 2.05204i) q^{8} +(-0.161782 - 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.53795	−	0.276019i	−1.79460	−	0.195175i	−0.850756	−	0.525561i	\(-0.823855\pi\)
							−0.943846	+	0.330387i	\(0.892821\pi\)
\(3\)	−0.647386	+	0.762162i	−0.373769	+	0.440034i
							\(5\)	−0.0920748	−	0.0699934i	−0.0411771	−	0.0313020i	0.584383	−	0.811478i	\(-0.301336\pi\)
−0.625560	+	0.780176i	\(0.715130\pi\)
\(7\)	0.00524223	−	0.0131570i	0.00198138	−	0.00497288i	−0.927983	−	0.372623i	\(-0.878458\pi\)
							0.929964	+	0.367650i	\(0.119838\pi\)
\(11\)	0.676236	−	0.312860i	0.203893	−	0.0943309i	−0.315285	−	0.948997i	\(-0.602100\pi\)
							0.519178	+	0.854666i	\(0.326238\pi\)
\(13\)	−0.486320	+	2.96642i	−0.134881	+	0.822736i	0.829556	+	0.558423i	\(0.188593\pi\)
							−0.964437	+	0.264313i	\(0.914855\pi\)
\(17\)	2.12270	+	5.32757i	0.514830	+	1.29213i	0.924948	+	0.380093i	\(0.124108\pi\)
							−0.410118	+	0.912032i	\(0.634513\pi\)
\(19\)	3.20593	+	4.72839i	0.735491	+	1.08477i	0.992956	+	0.118481i	\(0.0378023\pi\)
							−0.257465	+	0.966288i	\(0.582887\pi\)
\(23\)	−0.447639	+	8.25622i	−0.0933392	+	1.72154i	0.459436	+	0.888211i	\(0.348051\pi\)
							−0.552776	+	0.833330i	\(0.686431\pi\)
\(29\)	3.10832	−	0.338050i	0.577201	−	0.0627743i	0.185137	−	0.982713i	\(-0.440727\pi\)
							0.392063	+	0.919938i	\(0.371761\pi\)
\(31\)	−2.57450	+	3.79710i	−0.462393	+	0.681979i	−0.985382	−	0.170361i	\(-0.945507\pi\)
							0.522989	+	0.852340i	\(0.324817\pi\)
\(37\)	−4.76394	+	1.60516i	−0.783187	+	0.263887i	−0.682359	−	0.731017i	\(-0.739046\pi\)
							−0.100828	+	0.994904i	\(0.532149\pi\)
\(41\)	−0.347151	−	6.40283i	−0.0542159	−	0.999953i	−0.891453	−	0.453113i	\(-0.850313\pi\)
							0.837237	−	0.546840i	\(-0.184169\pi\)
\(43\)	−2.92854	−	1.35489i	−0.446599	−	0.206619i	0.183682	−	0.982986i	\(-0.441198\pi\)
							−0.630281	+	0.776367i	\(0.717060\pi\)
\(47\)	−2.19417	+	1.66796i	−0.320053	+	0.243298i	−0.752848	−	0.658195i	\(-0.771320\pi\)
							0.432795	+	0.901492i	\(0.357527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.4.1	✓	140
3.2	odd	2		531.2.i.b.181.5		140
59.15	even	29	inner	177.2.e.b.133.1	yes	140
177.74	odd	58		531.2.i.b.487.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.4.1	✓	140	1.1	even	1	trivial
177.2.e.b.133.1	yes	140	59.15	even	29	inner
531.2.i.b.181.5		140	3.2	odd	2
531.2.i.b.487.5		140	177.74	odd	58