Embedded newform 177.4.e.b.133.6

• Label: 177.4.e.b.133.6
• 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			133.6
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68795 + 0.183576i) q^{2} +(1.94216 + 2.28649i) q^{3} +(-4.99749 + 1.10003i) q^{4} +(3.49616 - 2.65771i) q^{5} +(-3.69801 - 3.50294i) q^{6} +(0.0781424 + 0.196123i) q^{7} +(21.1058 - 7.11136i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68795	+	0.183576i	−0.596780	+	0.0649038i	−0.401524	−	0.915848i	\(-0.631519\pi\)
							−0.195256	+	0.980752i	\(0.562554\pi\)
\(3\)	1.94216	+	2.28649i	0.373769	+	0.440034i
							\(5\)	3.49616	−	2.65771i	0.312706	−	0.237713i	−0.437035	−	0.899445i	\(-0.643971\pi\)
0.749741	+	0.661732i	\(0.230178\pi\)
\(7\)	0.0781424	+	0.196123i	0.00421930	+	0.0105896i	0.931071	−	0.364837i	\(-0.118875\pi\)
							−0.926852	+	0.375427i	\(0.877496\pi\)
\(11\)	21.6921	+	10.0358i	0.594584	+	0.275084i	0.694015	−	0.719961i	\(-0.255840\pi\)
							−0.0994310	+	0.995044i	\(0.531702\pi\)
\(13\)	0.213949	+	1.30503i	0.00456453	+	0.0278424i	0.989011	−	0.147845i	\(-0.0472336\pi\)
							−0.984446	+	0.175687i	\(0.943785\pi\)
\(17\)	−48.2163	+	121.014i	−0.687893	+	1.72648i	−0.000705979		1.00000i	\(0.500225\pi\)
							−0.687187	+	0.726481i	\(0.741155\pi\)
\(19\)	−10.2176	+	15.0698i	−0.123372	+	0.181960i	−0.884360	−	0.466806i	\(-0.845405\pi\)
							0.760987	+	0.648767i	\(0.224715\pi\)
\(23\)	−2.72912	−	50.3356i	−0.0247418	−	0.456335i	−0.984486	−	0.175461i	\(-0.943858\pi\)
							0.959745	−	0.280874i	\(-0.0906243\pi\)
\(29\)	−294.685	−	32.0489i	−1.88695	−	0.205218i	−0.909402	−	0.415918i	\(-0.863460\pi\)
							−0.977551	+	0.210700i	\(0.932426\pi\)
\(31\)	86.2936	+	127.273i	0.499961	+	0.737387i	0.991016	−	0.133741i	\(-0.0426991\pi\)
							−0.491056	+	0.871128i	\(0.663389\pi\)
\(37\)	358.654	+	120.845i	1.59358	+	0.536939i	0.969641	−	0.244534i	\(-0.0786350\pi\)
							0.623938	+	0.781474i	\(0.285532\pi\)
\(41\)	−23.3284	+	430.267i	−0.0888605	+	1.63894i	0.526132	+	0.850403i	\(0.323642\pi\)
							−0.614993	+	0.788533i	\(0.710841\pi\)
\(43\)	−394.172	+	182.363i	−1.39792	+	0.646748i	−0.966797	−	0.255544i	\(-0.917745\pi\)
							−0.431126	+	0.902292i	\(0.641883\pi\)
\(47\)	278.874	+	211.995i	0.865489	+	0.657928i	0.940700	−	0.339240i	\(-0.110170\pi\)
							−0.0752105	+	0.997168i	\(0.523963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.133.6	yes	420
59.4	even	29	inner	177.4.e.b.4.6	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.6	✓	420	59.4	even	29	inner
177.4.e.b.133.6	yes	420	1.1	even	1	trivial