Embedded newform 177.4.e.a.133.12

• Label: 177.4.e.a.133.12
• 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			133.12
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.a.4.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.41158 - 0.371031i) q^{2} +(-1.94216 - 2.28649i) q^{3} +(3.68824 - 0.811842i) q^{4} +(-7.61387 + 5.78791i) q^{5} +(-7.47419 - 7.07992i) q^{6} +(7.80642 + 19.5926i) q^{7} +(-13.7350 + 4.62785i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.41158	−	0.371031i	1.20618	−	0.131179i	0.517136	−	0.855903i	\(-0.326998\pi\)
							0.689039	+	0.724724i	\(0.258033\pi\)
\(3\)	−1.94216	−	2.28649i	−0.373769	−	0.440034i
							\(5\)	−7.61387	+	5.78791i	−0.681005	+	0.517687i	−0.887527	−	0.460755i	\(-0.847579\pi\)
0.206522	+	0.978442i	\(0.433785\pi\)
\(7\)	7.80642	+	19.5926i	0.421507	+	1.05790i	0.974434	+	0.224674i	\(0.0721316\pi\)
							−0.552927	+	0.833230i	\(0.686489\pi\)
\(11\)	14.1102	+	6.52809i	0.386763	+	0.178936i	0.603628	−	0.797266i	\(-0.293721\pi\)
							−0.216865	+	0.976202i	\(0.569583\pi\)
\(13\)	8.34196	+	50.8837i	0.177973	+	1.08558i	0.913124	+	0.407682i	\(0.133663\pi\)
							−0.735151	+	0.677903i	\(0.762889\pi\)
\(17\)	8.64645	−	21.7009i	0.123357	−	0.309603i	−0.854310	−	0.519765i	\(-0.826020\pi\)
							0.977667	+	0.210161i	\(0.0673990\pi\)
\(19\)	−17.9506	+	26.4751i	−0.216745	+	0.319674i	−0.920481	−	0.390786i	\(-0.872203\pi\)
							0.703737	+	0.710461i	\(0.251513\pi\)
\(23\)	0.433515	+	7.99572i	0.00393018	+	0.0724879i	0.999841	−	0.0178593i	\(-0.00568509\pi\)
							−0.995910	+	0.0903472i	\(0.971202\pi\)
\(29\)	19.5129	+	2.12216i	0.124947	+	0.0135888i	0.170379	−	0.985379i	\(-0.445501\pi\)
							−0.0454318	+	0.998967i	\(0.514466\pi\)
\(31\)	−78.5909	−	115.913i	−0.455333	−	0.671567i	0.528851	−	0.848715i	\(-0.322623\pi\)
							−0.984184	+	0.177148i	\(0.943313\pi\)
\(37\)	65.8331	+	22.1817i	0.292510	+	0.0985582i	0.461731	−	0.887020i	\(-0.347229\pi\)
							−0.169221	+	0.985578i	\(0.554125\pi\)
\(41\)	−8.26791	+	152.493i	−0.0314934	+	0.580862i	0.939616	+	0.342229i	\(0.111182\pi\)
							−0.971110	+	0.238633i	\(0.923301\pi\)
\(43\)	−23.6667	+	10.9494i	−0.0839335	+	0.0388318i	−0.461407	−	0.887189i	\(-0.652655\pi\)
							0.377473	+	0.926020i	\(0.376793\pi\)
\(47\)	−231.285	−	175.818i	−0.717795	−	0.545654i	0.181289	−	0.983430i	\(-0.441973\pi\)
							−0.899084	+	0.437776i	\(0.855766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.133.12	yes	420
59.4	even	29	inner	177.4.e.a.4.12	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.12	✓	420	59.4	even	29	inner
177.4.e.a.133.12	yes	420	1.1	even	1	trivial