Embedded newform 177.2.e.a.133.5

• Label: 177.2.e.a.133.5
• Level: $177$
• Weight: $2$
• Character: 177.133
• 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			133.5
Character	\(\chi\)	\(=\)	177.133
Dual form			177.2.e.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.15319 - 0.234174i) q^{2} +(0.647386 + 0.762162i) q^{3} +(2.62816 - 0.578503i) q^{4} +(-1.34628 + 1.02342i) q^{5} +(1.57243 + 1.48948i) q^{6} +(-0.734379 - 1.84315i) q^{7} +(1.41845 - 0.477932i) q^{8} +(-0.161782 + 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.15319	−	0.234174i	1.52254	−	0.165586i	0.691734	−	0.722152i	\(-0.256847\pi\)
							0.830803	+	0.556566i	\(0.187881\pi\)
\(3\)	0.647386	+	0.762162i	0.373769	+	0.440034i
							\(5\)	−1.34628	+	1.02342i	−0.602077	+	0.457687i	−0.861276	−	0.508137i	\(-0.830334\pi\)
0.259200	+	0.965824i	\(0.416541\pi\)
\(7\)	−0.734379	−	1.84315i	−0.277569	−	0.696646i	−0.999987	−	0.00518862i	\(-0.998348\pi\)
							0.722418	−	0.691457i	\(-0.243031\pi\)
\(11\)	0.997895	+	0.461676i	0.300877	+	0.139200i	0.564519	−	0.825420i	\(-0.309062\pi\)
							−0.263642	+	0.964621i	\(0.584924\pi\)
\(13\)	−0.907167	−	5.53347i	−0.251603	−	1.53471i	−0.746280	−	0.665633i	\(-0.768162\pi\)
							0.494677	−	0.869077i	\(-0.335286\pi\)
\(17\)	−0.674862	+	1.69377i	−0.163678	+	0.410801i	−0.987824	−	0.155577i	\(-0.950276\pi\)
							0.824146	+	0.566378i	\(0.191656\pi\)
\(19\)	−1.30465	+	1.92422i	−0.299308	+	0.441447i	−0.947370	−	0.320141i	\(-0.896270\pi\)
							0.648061	+	0.761588i	\(0.275580\pi\)
\(23\)	−0.198986	−	3.67009i	−0.0414915	−	0.765266i	−0.943059	−	0.332627i	\(-0.892065\pi\)
							0.901567	−	0.432639i	\(-0.142418\pi\)
\(29\)	8.77534	+	0.954376i	1.62954	+	0.177223i	0.876720	−	0.481002i	\(-0.159727\pi\)
							0.752821	+	0.658225i	\(0.228693\pi\)
\(31\)	0.663059	+	0.977939i	0.119089	+	0.175643i	0.882561	−	0.470198i	\(-0.155817\pi\)
							−0.763472	+	0.645841i	\(0.776507\pi\)
\(37\)	−3.77049	−	1.27043i	−0.619865	−	0.208857i	−0.00819045	−	0.999966i	\(-0.502607\pi\)
							−0.611674	+	0.791110i	\(0.709504\pi\)
\(41\)	−0.376397	+	6.94224i	−0.0587834	+	1.08420i	0.809178	+	0.587564i	\(0.199913\pi\)
							−0.867961	+	0.496632i	\(0.834570\pi\)
\(43\)	3.49435	−	1.61666i	0.532884	−	0.246538i	−0.134945	−	0.990853i	\(-0.543086\pi\)
							0.667829	+	0.744315i	\(0.267224\pi\)
\(47\)	−0.973724	−	0.740206i	−0.142032	−	0.107970i	0.531745	−	0.846904i	\(-0.321536\pi\)
							−0.673778	+	0.738934i	\(0.735329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.133.5	yes	140
3.2	odd	2		531.2.i.c.487.1		140
59.4	even	29	inner	177.2.e.a.4.5	✓	140
177.122	odd	58		531.2.i.c.181.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.4.5	✓	140	59.4	even	29	inner
177.2.e.a.133.5	yes	140	1.1	even	1	trivial
531.2.i.c.181.1		140	177.122	odd	58
531.2.i.c.487.1		140	3.2	odd	2