Embedded newform 177.2.e.a.28.4

• Label: 177.2.e.a.28.4
• 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			28.4
Character	\(\chi\)	\(=\)	177.28
Dual form			177.2.e.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.655840 + 1.23705i) q^{2} +(-0.725995 - 0.687699i) q^{3} +(0.0222185 - 0.0327698i) q^{4} +(2.32817 + 0.512469i) q^{5} +(0.374579 - 1.34911i) q^{6} +(-1.36308 - 1.03619i) q^{7} +(2.83899 + 0.308758i) q^{8} +(0.0541389 + 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.655840	+	1.23705i	0.463749	+	0.874723i	0.999539	+	0.0303719i	\(0.00966915\pi\)
							−0.535790	+	0.844351i	\(0.679986\pi\)
\(3\)	−0.725995	−	0.687699i	−0.419154	−	0.397043i
							\(5\)	2.32817	+	0.512469i	1.04119	+	0.229183i	0.702476	−	0.711707i	\(-0.252078\pi\)
0.338712	+	0.940890i	\(0.390009\pi\)
\(7\)	−1.36308	−	1.03619i	−0.515198	−	0.391643i	0.315018	−	0.949086i	\(-0.397990\pi\)
							−0.830215	+	0.557443i	\(0.811783\pi\)
\(11\)	0.846770	−	2.12523i	0.255311	−	0.640782i	−0.744324	−	0.667819i	\(-0.767228\pi\)
							0.999634	+	0.0270373i	\(0.00860728\pi\)
\(13\)	−0.224453	+	4.13980i	−0.0622521	+	1.14817i	0.785877	+	0.618383i	\(0.212212\pi\)
							−0.848129	+	0.529790i	\(0.822271\pi\)
\(17\)	−2.30690	+	1.75366i	−0.559505	+	0.425325i	−0.846330	−	0.532658i	\(-0.821193\pi\)
							0.286825	+	0.957983i	\(0.407400\pi\)
\(19\)	−1.04728	−	0.352870i	−0.240262	−	0.0809538i	0.196594	−	0.980485i	\(-0.437012\pi\)
							−0.436857	+	0.899531i	\(0.643908\pi\)
\(23\)	−4.30312	−	2.58910i	−0.897264	−	0.539866i	−0.00939142	−	0.999956i	\(-0.502989\pi\)
							−0.887872	+	0.460090i	\(0.847817\pi\)
\(29\)	2.49387	−	4.70395i	0.463101	−	0.873501i	−0.536462	−	0.843925i	\(-0.680239\pi\)
							0.999563	−	0.0295763i	\(-0.00941581\pi\)
\(31\)	−2.74214	+	0.923934i	−0.492502	+	0.165943i	−0.554584	−	0.832128i	\(-0.687123\pi\)
							0.0620817	+	0.998071i	\(0.480226\pi\)
\(37\)	−9.84258	+	1.07044i	−1.61811	+	0.175980i	−0.871884	−	0.489713i	\(-0.837102\pi\)
							−0.746226	+	0.665693i	\(0.768136\pi\)
\(41\)	−3.19256	+	1.92090i	−0.498594	+	0.299994i	−0.742548	−	0.669793i	\(-0.766383\pi\)
							0.243954	+	0.969787i	\(0.421555\pi\)
\(43\)	0.835041	+	2.09579i	0.127343	+	0.319606i	0.978800	−	0.204818i	\(-0.0656602\pi\)
							−0.851458	+	0.524423i	\(0.824281\pi\)
\(47\)	7.54956	−	1.66178i	1.10122	−	0.242396i	0.373060	−	0.927807i	\(-0.378309\pi\)
							0.728156	+	0.685411i	\(0.240378\pi\)

(See \(a_n\) instead)

Twists

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

    

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