Embedded newform 177.4.e.a.4.12

• Label: 177.4.e.a.4.12
• Level: $177$
• Weight: $4$
• Character: 177.4
• 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			4.12
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.12

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.12	✓	420	1.1	even	1	trivial
177.4.e.a.133.12	yes	420	59.15	even	29	inner