Embedded newform 76.4.f.a.27.14

• Label: 76.4.f.a.27.14
• Level: $76$
• Weight: $4$
• Character: 76.27
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.14
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0248348 - 2.82832i) q^{2} +(4.46255 - 7.72937i) q^{3} +(-7.99877 + 0.140481i) q^{4} +(-6.17128 + 10.6890i) q^{5} +(-21.9719 - 12.4296i) q^{6} -29.2047i q^{7} +(0.595974 + 22.6196i) q^{8} +(-26.3288 - 45.6028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0248348	−	2.82832i	−0.00878042	−	0.999961i
							\(3\)	4.46255	−	7.72937i	0.858819	−	1.48752i	−0.0142377	−	0.999899i	\(-0.504532\pi\)
0.873056	−	0.487619i	\(-0.162135\pi\)
\(5\)	−6.17128	+	10.6890i	−0.551976	+	0.956051i	0.446156	+	0.894955i	\(0.352793\pi\)
							−0.998132	+	0.0610956i	\(0.980541\pi\)
\(7\)		−	29.2047i		−	1.57691i	−0.615096	−	0.788453i	\(-0.710883\pi\)
							0.615096	−	0.788453i	\(-0.289117\pi\)
\(11\)			34.0259i			0.932654i	0.884613	+	0.466327i	\(0.154423\pi\)
							−0.884613	+	0.466327i	\(0.845577\pi\)
\(13\)	42.7968	−	24.7087i	0.913054	−	0.527152i	0.0316413	−	0.999499i	\(-0.489927\pi\)
							0.881412	+	0.472348i	\(0.156593\pi\)
\(17\)	0.493482	−	0.854735i	0.00704040	−	0.0121943i	−0.862484	−	0.506085i	\(-0.831092\pi\)
							0.869524	+	0.493890i	\(0.164426\pi\)
\(19\)	−70.4393	−	43.5580i	−0.850521	−	0.525942i
							\(23\)	175.004	−	101.039i	1.58656	−	0.916000i	0.592690	−	0.805431i	\(-0.298066\pi\)
0.993868	−	0.110569i	\(-0.0352674\pi\)
\(29\)	−16.9232	+	9.77059i	−0.108364	+	0.0625639i	−0.553202	−	0.833047i	\(-0.686594\pi\)
							0.444839	+	0.895611i	\(0.353261\pi\)
\(31\)	−49.1996			−0.285048			−0.142524	−	0.989791i	\(-0.545522\pi\)
							−0.142524	+	0.989791i	\(0.545522\pi\)
\(37\)		−	121.069i		−	0.537935i	−0.963149	−	0.268968i	\(-0.913318\pi\)
							0.963149	−	0.268968i	\(-0.0866824\pi\)
\(41\)	119.127	+	68.7779i	0.453768	+	0.261983i	0.709420	−	0.704786i	\(-0.248957\pi\)
							−0.255652	+	0.966769i	\(0.582290\pi\)
\(43\)	133.019	+	76.7986i	0.471749	+	0.272364i	0.716972	−	0.697102i	\(-0.245528\pi\)
							−0.245223	+	0.969467i	\(0.578861\pi\)
\(47\)	289.588	−	167.193i	0.898738	−	0.518887i	0.0219476	−	0.999759i	\(-0.493013\pi\)
							0.876791	+	0.480872i	\(0.159680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.14	yes	56
4.3	odd	2	inner	76.4.f.a.27.5	✓	56
19.12	odd	6	inner	76.4.f.a.31.5	yes	56
76.31	even	6	inner	76.4.f.a.31.14	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.5	✓	56	4.3	odd	2	inner
76.4.f.a.27.14	yes	56	1.1	even	1	trivial
76.4.f.a.31.5	yes	56	19.12	odd	6	inner
76.4.f.a.31.14	yes	56	76.31	even	6	inner