Embedded newform 77.3.p.a.5.5

• Label: 77.3.p.a.5.5
• Level: $77$
• Weight: $3$
• Character: 77.5
• Analytic conductor: $2.098$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	77.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.09809803557\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	77.5
Dual form			77.3.p.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85255 - 0.393773i) q^{2} +(-4.21020 - 0.442510i) q^{3} +(-0.377280 - 0.167976i) q^{4} +(1.58739 - 1.42930i) q^{5} +(7.62537 + 2.47763i) q^{6} +(4.15912 + 5.63043i) q^{7} +(6.76171 + 4.91267i) q^{8} +(8.72662 + 1.85490i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85255	−	0.393773i	−0.926277	−	0.196886i	−0.280010	−	0.959997i	\(-0.590338\pi\)
							−0.646268	+	0.763111i	\(0.723671\pi\)
\(3\)	−4.21020	−	0.442510i	−1.40340	−	0.147503i	−0.627626	−	0.778515i	\(-0.715973\pi\)
							−0.775773	+	0.631012i	\(0.782640\pi\)
\(5\)	1.58739	−	1.42930i	0.317479	−	0.285859i	−0.494948	−	0.868923i	\(-0.664813\pi\)
							0.812427	+	0.583064i	\(0.198146\pi\)
\(7\)	4.15912	+	5.63043i	0.594161	+	0.804346i
							\(11\)	−4.85155	+	9.87231i	−0.441050	+	0.897482i
\(13\)	10.9211	−	3.54847i	0.840082	−	0.272959i								0.142796	−	0.989752i	\(-0.454391\pi\)
							0.697286	+	0.716793i	\(0.254391\pi\)
\(17\)	−2.48396	−	11.6861i	−0.146115	−	0.687418i	−0.988829	−	0.149056i	\(-0.952377\pi\)
							0.842713	−	0.538362i	\(-0.180957\pi\)
\(19\)	14.0244	+	31.4993i	0.738126	+	1.65786i	0.753045	+	0.657969i	\(0.228584\pi\)
							−0.0149196	+	0.999889i	\(0.504749\pi\)
\(23\)	4.74153	−	8.21257i	0.206153	−	0.357068i	−0.744346	−	0.667794i	\(-0.767239\pi\)
							0.950500	+	0.310726i	\(0.100572\pi\)
\(29\)	−33.5230	+	24.3559i	−1.15597	+	0.839859i	−0.989263	−	0.146148i	\(-0.953312\pi\)
							−0.166704	+	0.986007i	\(0.553312\pi\)
\(31\)	15.3229	+	13.7968i	0.494287	+	0.445058i	0.878188	−	0.478317i	\(-0.158753\pi\)
							−0.383901	+	0.923374i	\(0.625419\pi\)
\(37\)	3.74060	+	35.5894i	0.101097	+	0.961876i	0.921050	+	0.389445i	\(0.127333\pi\)
							−0.819953	+	0.572432i	\(0.806000\pi\)
\(41\)	21.9979	−	30.2775i	0.536534	−	0.738476i	−0.451574	−	0.892233i	\(-0.649137\pi\)
							0.988109	+	0.153758i	\(0.0491375\pi\)
\(43\)	18.9644			0.441033			0.220516	−	0.975383i	\(-0.429226\pi\)
							0.220516	+	0.975383i	\(0.429226\pi\)
\(47\)	2.56561	+	5.76246i	0.0545875	+	0.122606i	0.938780	−	0.344517i	\(-0.111957\pi\)
							−0.884193	+	0.467123i	\(0.845291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.3.p.a.5.5	✓	112
7.3	odd	6	inner	77.3.p.a.38.10	yes	112
11.9	even	5	inner	77.3.p.a.75.10	yes	112
77.31	odd	30	inner	77.3.p.a.31.5	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.3.p.a.5.5	✓	112	1.1	even	1	trivial
77.3.p.a.31.5	yes	112	77.31	odd	30	inner
77.3.p.a.38.10	yes	112	7.3	odd	6	inner
77.3.p.a.75.10	yes	112	11.9	even	5	inner