Embedded newform 177.2.e.a.28.2

• Label: 177.2.e.a.28.2
• Level: $177$
• Weight: $2$
• Character: 177.28
• 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			28.2
Character	\(\chi\)	\(=\)	177.28
Dual form			177.2.e.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.454038 - 0.856407i) q^{2} +(-0.725995 - 0.687699i) q^{3} +(0.595092 - 0.877695i) q^{4} +(3.60206 + 0.792874i) q^{5} +(-0.259321 + 0.933989i) q^{6} +(1.56865 + 1.19245i) q^{7} +(-2.94914 - 0.320738i) q^{8} +(0.0541389 + 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{10}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.454038	−	0.856407i	−0.321054	−	0.605571i	0.669482	−	0.742828i	\(-0.266516\pi\)
							−0.990535	+	0.137257i	\(0.956171\pi\)
\(3\)	−0.725995	−	0.687699i	−0.419154	−	0.397043i
							\(5\)	3.60206	+	0.792874i	1.61089	+	0.354584i	0.927018	−	0.375017i	\(-0.122363\pi\)
0.683873	+	0.729601i	\(0.260294\pi\)
\(7\)	1.56865	+	1.19245i	0.592893	+	0.450706i	0.858098	−	0.513487i	\(-0.171646\pi\)
							−0.265205	+	0.964192i	\(0.585439\pi\)
\(11\)	−0.731304	+	1.83543i	−0.220496	+	0.553404i	−0.997059	−	0.0766358i	\(-0.975582\pi\)
							0.776563	+	0.630040i	\(0.216961\pi\)
\(13\)	0.326231	−	6.01697i	0.0904801	−	1.66881i	−0.503248	−	0.864142i	\(-0.667862\pi\)
							0.593728	−	0.804665i	\(-0.297655\pi\)
\(17\)	−1.98052	+	1.50555i	−0.480347	+	0.365150i	−0.817150	−	0.576425i	\(-0.804447\pi\)
							0.336803	+	0.941575i	\(0.390654\pi\)
\(19\)	−0.510808	−	0.172111i	−0.117187	−	0.0394850i	0.260097	−	0.965583i	\(-0.416245\pi\)
							−0.377284	+	0.926097i	\(0.623142\pi\)
\(23\)	−2.30855	−	1.38901i	−0.481366	−	0.289628i	0.254115	−	0.967174i	\(-0.418216\pi\)
							−0.735481	+	0.677546i	\(0.763044\pi\)
\(29\)	−4.68698	+	8.84059i	−0.870351	+	1.64166i	−0.108319	+	0.994116i	\(0.534547\pi\)
							−0.762032	+	0.647540i	\(0.775798\pi\)
\(31\)	−6.98124	+	2.35225i	−1.25387	+	0.422477i	−0.866374	−	0.499395i	\(-0.833556\pi\)
							−0.387493	+	0.921873i	\(0.626659\pi\)
\(37\)	8.07738	−	0.878468i	1.32791	−	0.144419i	0.583486	−	0.812123i	\(-0.301688\pi\)
							0.744427	+	0.667704i	\(0.232723\pi\)
\(41\)	4.71579	−	2.83740i	0.736482	−	0.443127i	−0.0972751	−	0.995258i	\(-0.531013\pi\)
							0.833758	+	0.552131i	\(0.186185\pi\)
\(43\)	1.43226	+	3.59471i	0.218418	+	0.548189i	0.996830	−	0.0795660i	\(-0.0253534\pi\)
							−0.778411	+	0.627755i	\(0.783974\pi\)
\(47\)	−1.10146	+	0.242450i	−0.160665	+	0.0353649i	−0.294574	−	0.955629i	\(-0.595178\pi\)
							0.133910	+	0.990994i	\(0.457247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.28.2	yes	140
3.2	odd	2		531.2.i.c.28.4		140
59.19	even	29	inner	177.2.e.a.19.2	✓	140
177.137	odd	58		531.2.i.c.19.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.19.2	✓	140	59.19	even	29	inner
177.2.e.a.28.2	yes	140	1.1	even	1	trivial
531.2.i.c.19.4		140	177.137	odd	58
531.2.i.c.28.4		140	3.2	odd	2