Embedded newform 78.5.j.a.17.10

• Label: 78.5.j.a.17.10
• Level: $78$
• Weight: $5$
• Character: 78.17
• Analytic conductor: $8.063$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	78.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.06285712054\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.10
Character	\(\chi\)	\(=\)	78.17
Dual form			78.5.j.a.23.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41421 - 2.44949i) q^{2} +(-8.53538 - 2.85433i) q^{3} +(-4.00000 - 6.92820i) q^{4} -25.8074 q^{5} +(-19.0625 + 16.8707i) q^{6} +(47.1333 - 27.2124i) q^{7} -22.6274 q^{8} +(64.7056 + 48.7256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 144 q^{4} + 126 q^{7} + 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41421	−	2.44949i	0.353553	−	0.612372i
							\(3\)	−8.53538	−	2.85433i	−0.948376	−	0.317148i
\(5\)	−25.8074			−1.03230										−0.516148	−	0.856499i	\(-0.672635\pi\)
							−0.516148	+	0.856499i	\(0.672635\pi\)
\(7\)	47.1333	−	27.2124i	0.961905	−	0.555356i	0.0651460	−	0.997876i	\(-0.479249\pi\)
							0.896759	+	0.442520i	\(0.145915\pi\)
\(11\)	−120.177	+	208.153i	−0.993198	+	1.72027i	−0.395759	+	0.918355i	\(0.629518\pi\)
							−0.597439	+	0.801914i	\(0.703815\pi\)
\(13\)	−49.2878	+	161.653i	−0.291644	+	0.956527i
							\(17\)	238.563	−	137.735i	0.825479	−	0.476590i	−0.0268233	−	0.999640i	\(-0.508539\pi\)
0.852302	+	0.523050i	\(0.175206\pi\)
\(19\)	−154.747	+	89.3434i	−0.428663	+	0.247489i	−0.698777	−	0.715340i	\(-0.746272\pi\)
							0.270114	+	0.962828i	\(0.412939\pi\)
\(23\)	−398.379	−	230.004i	−0.753080	−	0.434791i	0.0737257	−	0.997279i	\(-0.476511\pi\)
							−0.826806	+	0.562488i	\(0.809844\pi\)
\(29\)	30.3129	+	17.5012i	0.0360439	+	0.0208100i	0.517914	−	0.855433i	\(-0.326709\pi\)
							−0.481870	+	0.876243i	\(0.660042\pi\)
\(31\)			955.516i			0.994293i	0.867666	+	0.497147i	\(0.165619\pi\)
							−0.867666	+	0.497147i	\(0.834381\pi\)
\(37\)	−1269.59	−	732.999i	−0.927385	−	0.535426i	−0.0414018	−	0.999143i	\(-0.513182\pi\)
							−0.885984	+	0.463716i	\(0.846516\pi\)
\(41\)	−1476.96	+	2558.17i	−0.878619	+	1.52181i	−0.0257615	+	0.999668i	\(0.508201\pi\)
							−0.852857	+	0.522144i	\(0.825132\pi\)
\(43\)	−1258.33	−	2179.49i	−0.680546	−	1.17874i	−0.974815	−	0.223017i	\(-0.928410\pi\)
							0.294269	−	0.955723i	\(-0.404924\pi\)
\(47\)	1497.81			0.678047			0.339024	−	0.940778i	\(-0.389903\pi\)
							0.339024	+	0.940778i	\(0.389903\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.5.j.a.17.10	yes	36
3.2	odd	2	inner	78.5.j.a.17.7	✓	36
13.10	even	6	inner	78.5.j.a.23.7	yes	36
39.23	odd	6	inner	78.5.j.a.23.10	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.5.j.a.17.7	✓	36	3.2	odd	2	inner
78.5.j.a.17.10	yes	36	1.1	even	1	trivial
78.5.j.a.23.7	yes	36	13.10	even	6	inner
78.5.j.a.23.10	yes	36	39.23	odd	6	inner