Embedded newform 79.5.d.a.24.16

• Label: 79.5.d.a.24.16
• Level: $79$
• Weight: $5$
• Character: 79.24
• Analytic conductor: $8.166$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	79.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.16622708362\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			24.16
Character	\(\chi\)	\(=\)	79.24
Dual form			79.5.d.a.56.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.697858 - 1.20873i) q^{2} +(-8.47508 - 4.89309i) q^{3} +(7.02599 + 12.1694i) q^{4} +(-0.519176 - 0.899239i) q^{5} +(-11.8288 + 6.82937i) q^{6} +(36.3927 - 21.0114i) q^{7} +41.9440 q^{8} +(7.38469 + 12.7907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 2 q^{2} + 6 q^{3} - 234 q^{4} - q^{5} - 3 q^{6} + 120 q^{7} + 112 q^{8} + 640 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.697858	−	1.20873i	0.174465	−	0.302182i	−0.765511	−	0.643422i	\(-0.777514\pi\)
							0.939976	+	0.341241i	\(0.110847\pi\)
\(3\)	−8.47508	−	4.89309i	−0.941676	−	0.543677i	−0.0511907	−	0.998689i	\(-0.516302\pi\)
							−0.890485	+	0.455012i	\(0.849635\pi\)
\(5\)	−0.519176	−	0.899239i	−0.0207670	−	0.0359695i	0.855455	−	0.517877i	\(-0.173277\pi\)
							−0.876222	+	0.481907i	\(0.839944\pi\)
\(7\)	36.3927	−	21.0114i	0.742709	−	0.428803i	−0.0803444	−	0.996767i	\(-0.525602\pi\)
							0.823053	+	0.567964i	\(0.192269\pi\)
\(11\)	−84.8012	−	146.880i	−0.700836	−	1.21388i	−0.968173	−	0.250281i	\(-0.919477\pi\)
							0.267337	−	0.963603i	\(-0.413856\pi\)
\(13\)	30.2792	−	52.4450i	0.179167	−	0.310326i	−0.762429	−	0.647072i	\(-0.775993\pi\)
							0.941595	+	0.336747i	\(0.109327\pi\)
\(17\)		−	353.785i		−	1.22417i	−0.790793	−	0.612084i	\(-0.790331\pi\)
							0.790793	−	0.612084i	\(-0.209669\pi\)
\(19\)	−112.447	−	194.765i	−0.311489	−	0.539514i	0.667196	−	0.744882i	\(-0.267494\pi\)
							−0.978685	+	0.205368i	\(0.934161\pi\)
\(23\)	−24.4539	−	42.3555i	−0.0462267	−	0.0800670i	0.841986	−	0.539499i	\(-0.181386\pi\)
							−0.888213	+	0.459432i	\(0.848053\pi\)
\(29\)	992.352	−	572.935i	1.17997	−	0.681254i	0.223960	−	0.974598i	\(-0.428101\pi\)
							0.956007	+	0.293344i	\(0.0947681\pi\)
\(31\)	115.967	+	200.862i	0.120674	+	0.209013i	0.920034	−	0.391840i	\(-0.128161\pi\)
							−0.799360	+	0.600853i	\(0.794828\pi\)
\(37\)	−46.0706	−	26.5989i	−0.0336527	−	0.0194294i	0.483079	−	0.875577i	\(-0.339518\pi\)
							−0.516732	+	0.856147i	\(0.672852\pi\)
\(41\)		−	1486.50i		−	0.884295i	−0.896942	−	0.442147i	\(-0.854217\pi\)
							0.896942	−	0.442147i	\(-0.145783\pi\)
\(43\)	1406.80	+	812.217i	0.760844	+	0.439274i	0.829599	−	0.558360i	\(-0.188569\pi\)
							−0.0687547	+	0.997634i	\(0.521903\pi\)
\(47\)	−2982.46	+	1721.92i	−1.35014	+	0.779504i	−0.988269	−	0.152724i	\(-0.951195\pi\)
							−0.361871	+	0.932228i	\(0.617862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	79.5.d.a.24.16	✓	52
79.56	odd	6	inner	79.5.d.a.56.16	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
79.5.d.a.24.16	✓	52	1.1	even	1	trivial
79.5.d.a.56.16	yes	52	79.56	odd	6	inner