Embedded newform 177.4.e.b.133.1

• Label: 177.4.e.b.133.1
• Level: $177$
• Weight: $4$
• Character: 177.133
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			133.1
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.48057 + 0.596048i) q^{2} +(1.94216 + 2.28649i) q^{3} +(21.8684 - 4.81360i) q^{4} +(0.769914 - 0.585273i) q^{5} +(-12.0070 - 11.3736i) q^{6} +(4.65584 + 11.6853i) q^{7} +(-75.1876 + 25.3337i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.48057	+	0.596048i	−1.93767	+	0.210735i	−0.993074	−	0.117491i	\(-0.962515\pi\)
							−0.944599	+	0.328226i	\(0.893549\pi\)
\(3\)	1.94216	+	2.28649i	0.373769	+	0.440034i
							\(5\)	0.769914	−	0.585273i	0.0688632	−	0.0523484i	−0.570176	−	0.821522i	\(-0.693125\pi\)
0.639040	+	0.769174i	\(0.279332\pi\)
\(7\)	4.65584	+	11.6853i	0.251392	+	0.630945i	0.999466	−	0.0326693i	\(-0.0104008\pi\)
							−0.748075	+	0.663615i	\(0.769022\pi\)
\(11\)	−2.09487	−	0.969189i	−0.0574206	−	0.0265656i	0.390969	−	0.920404i	\(-0.372140\pi\)
							−0.448389	+	0.893838i	\(0.648002\pi\)
\(13\)	6.63504	+	40.4719i	0.141556	+	0.863453i	0.958155	+	0.286249i	\(0.0924083\pi\)
							−0.816599	+	0.577205i	\(0.804143\pi\)
\(17\)	1.58645	−	3.98169i	0.0226336	−	0.0568060i	−0.917226	−	0.398368i	\(-0.869577\pi\)
							0.939859	+	0.341562i	\(0.110956\pi\)
\(19\)	72.2438	−	106.552i	0.872309	−	1.28656i	−0.0846725	−	0.996409i	\(-0.526984\pi\)
							0.956981	−	0.290150i	\(-0.0937052\pi\)
\(23\)	3.74368	+	69.0482i	0.0339396	+	0.625980i	0.965182	+	0.261579i	\(0.0842433\pi\)
							−0.931242	+	0.364401i	\(0.881274\pi\)
\(29\)	−44.5628	−	4.84649i	−0.285348	−	0.0310335i	−0.0356743	−	0.999363i	\(-0.511358\pi\)
							−0.249674	+	0.968330i	\(0.580323\pi\)
\(31\)	118.127	+	174.224i	0.684393	+	1.00940i	0.998203	+	0.0599181i	\(0.0190840\pi\)
							−0.313811	+	0.949486i	\(0.601606\pi\)
\(37\)	−284.864	−	95.9818i	−1.26571	−	0.426468i	−0.395190	−	0.918599i	\(-0.629321\pi\)
							−0.870521	+	0.492132i	\(0.836218\pi\)
\(41\)	−13.4264	+	247.635i	−0.0511427	+	0.943272i	0.854559	+	0.519355i	\(0.173828\pi\)
							−0.905701	+	0.423916i	\(0.860655\pi\)
\(43\)	−358.528	+	165.873i	−1.27151	+	0.588264i	−0.935411	−	0.353561i	\(-0.884971\pi\)
							−0.336101	+	0.941826i	\(0.609108\pi\)
\(47\)	65.2556	+	49.6061i	0.202522	+	0.153953i	0.701519	−	0.712650i	\(-0.252505\pi\)
							−0.498998	+	0.866603i	\(0.666298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.133.1	yes	420
59.4	even	29	inner	177.4.e.b.4.1	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.1	✓	420	59.4	even	29	inner
177.4.e.b.133.1	yes	420	1.1	even	1	trivial