Embedded newform 76.6.f.a.27.10

• Label: 76.6.f.a.27.10
• Level: $76$
• Weight: $6$
• Character: 76.27
• Analytic conductor: $12.189$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1891703058\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.10
Character	\(\chi\)	\(=\)	76.27
Dual form			76.6.f.a.31.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.70334 + 3.14303i) q^{2} +(7.75035 - 13.4240i) q^{3} +(12.2427 - 29.5654i) q^{4} +(-8.78700 + 15.2195i) q^{5} +(5.73951 + 87.4972i) q^{6} +77.0445i q^{7} +(35.3433 + 177.535i) q^{8} +(1.36406 + 2.36263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.70334	+	3.14303i	−0.831440	+	0.555614i
							\(3\)	7.75035	−	13.4240i	0.497185	−	0.861150i	−0.502809	−	0.864397i	\(-0.667700\pi\)
0.999995	+	0.00324702i	\(0.00103356\pi\)
\(5\)	−8.78700	+	15.2195i	−0.157187	+	0.272255i	−0.933853	−	0.357657i	\(-0.883576\pi\)
							0.776666	+	0.629912i	\(0.216909\pi\)
\(7\)			77.0445i			0.594287i	0.954833	+	0.297144i	\(0.0960340\pi\)
							−0.954833	+	0.297144i	\(0.903966\pi\)
\(11\)			67.5657i			0.168362i	0.996450	+	0.0841811i	\(0.0268274\pi\)
							−0.996450	+	0.0841811i	\(0.973173\pi\)
\(13\)	538.434	−	310.865i	0.883638	−	0.510168i	0.0117816	−	0.999931i	\(-0.496250\pi\)
							0.871856	+	0.489762i	\(0.162916\pi\)
\(17\)	1125.58	−	1949.56i	0.944612	−	1.63612i	0.188087	−	0.982152i	\(-0.439771\pi\)
							0.756526	−	0.653964i	\(-0.226895\pi\)
\(19\)	−344.019	+	1535.50i	−0.218624	+	0.975809i
							\(23\)	1743.96	−	1006.88i	0.687412	−	0.396877i	−0.115230	−	0.993339i	\(-0.536760\pi\)
0.802642	+	0.596461i	\(0.203427\pi\)
\(29\)	4570.26	−	2638.64i	1.00913	−	0.582620i	0.0981919	−	0.995167i	\(-0.468694\pi\)
							0.910936	+	0.412547i	\(0.135361\pi\)
\(31\)	−1021.36			−0.190887			−0.0954435	−	0.995435i	\(-0.530427\pi\)
							−0.0954435	+	0.995435i	\(0.530427\pi\)
\(37\)		−	4516.66i		−	0.542391i	−0.962524	−	0.271196i	\(-0.912581\pi\)
							0.962524	−	0.271196i	\(-0.0874190\pi\)
\(41\)	5424.50	+	3131.83i	0.503964	+	0.290964i	0.730349	−	0.683074i	\(-0.239357\pi\)
							−0.226385	+	0.974038i	\(0.572691\pi\)
\(43\)	−7439.38	−	4295.13i	−0.613572	−	0.354246i	0.160790	−	0.986989i	\(-0.448596\pi\)
							−0.774362	+	0.632743i	\(0.781929\pi\)
\(47\)	−7786.40	+	4495.48i	−0.514152	+	0.296846i	−0.734539	−	0.678567i	\(-0.762601\pi\)
							0.220386	+	0.975413i	\(0.429268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.27.10	✓	96
4.3	odd	2	inner	76.6.f.a.27.26	yes	96
19.12	odd	6	inner	76.6.f.a.31.26	yes	96
76.31	even	6	inner	76.6.f.a.31.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.10	✓	96	1.1	even	1	trivial
76.6.f.a.27.26	yes	96	4.3	odd	2	inner
76.6.f.a.31.10	yes	96	76.31	even	6	inner
76.6.f.a.31.26	yes	96	19.12	odd	6	inner