Embedded newform 77.3.p.a.5.3

• Label: 77.3.p.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	77.5
Dual form			77.3.p.a.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.88156 - 0.612495i) q^{2} +(-0.771458 - 0.0810835i) q^{3} +(4.27406 + 1.90294i) q^{4} +(3.48954 - 3.14199i) q^{5} +(2.17334 + 0.706161i) q^{6} +(-6.97684 + 0.568896i) q^{7} +(-1.61719 - 1.17496i) q^{8} +(-8.21476 - 1.74610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.88156	−	0.612495i	−1.44078	−	0.306247i	−0.579748	−	0.814796i	\(-0.696849\pi\)
							−0.861033	+	0.508549i	\(0.830182\pi\)
\(3\)	−0.771458	−	0.0810835i	−0.257153	−	0.0270278i	−0.0249251	−	0.999689i	\(-0.507935\pi\)
							−0.232227	+	0.972661i	\(0.574601\pi\)
\(5\)	3.48954	−	3.14199i	0.697907	−	0.628398i	−0.241821	−	0.970321i	\(-0.577745\pi\)
							0.939728	+	0.341922i	\(0.111078\pi\)
\(7\)	−6.97684	+	0.568896i	−0.996692	+	0.0812708i
							\(11\)	−9.06979	−	6.22406i	−0.824526	−	0.565824i
\(13\)	−14.7758	+	4.80094i	−1.13660	+	0.369303i								−0.816080	−	0.577939i	\(-0.803857\pi\)
							−0.320518	+	0.947242i	\(0.603857\pi\)
\(17\)	4.18621	+	19.6946i	0.246248	+	1.15850i	0.911311	+	0.411718i	\(0.135071\pi\)
							−0.665064	+	0.746787i	\(0.731596\pi\)
\(19\)	−9.73884	−	21.8738i	−0.512570	−	1.15125i	−0.965665	−	0.259790i	\(-0.916347\pi\)
							0.453095	−	0.891462i	\(-0.350320\pi\)
\(23\)	15.9629	−	27.6485i	0.694038	−	1.20211i	−0.276465	−	0.961024i	\(-0.589163\pi\)
							0.970504	−	0.241086i	\(-0.0775036\pi\)
\(29\)	−5.51080	+	4.00383i	−0.190028	+	0.138063i	−0.678731	−	0.734387i	\(-0.737470\pi\)
							0.488704	+	0.872450i	\(0.337470\pi\)
\(31\)	−15.4859	−	13.9436i	−0.499545	−	0.449793i	0.380432	−	0.924809i	\(-0.375775\pi\)
							−0.879977	+	0.475017i	\(0.842442\pi\)
\(37\)	2.85597	+	27.1727i	0.0771884	+	0.734398i	0.962844	+	0.270057i	\(0.0870424\pi\)
							−0.885656	+	0.464342i	\(0.846291\pi\)
\(41\)	23.4987	−	32.3432i	0.573139	−	0.788859i	−0.419783	−	0.907625i	\(-0.637894\pi\)
							0.992922	+	0.118766i	\(0.0378938\pi\)
\(43\)	−34.6158			−0.805020			−0.402510	−	0.915416i	\(-0.631862\pi\)
							−0.402510	+	0.915416i	\(0.631862\pi\)
\(47\)	−13.9108	−	31.2442i	−0.295975	−	0.664771i	0.702945	−	0.711245i	\(-0.251868\pi\)
							−0.998919	+	0.0464741i	\(0.985202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.3.p.a.5.3	✓	112
7.3	odd	6	inner	77.3.p.a.38.12	yes	112
11.9	even	5	inner	77.3.p.a.75.12	yes	112
77.31	odd	30	inner	77.3.p.a.31.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.3.p.a.5.3	✓	112	1.1	even	1	trivial
77.3.p.a.31.3	yes	112	77.31	odd	30	inner
77.3.p.a.38.12	yes	112	7.3	odd	6	inner
77.3.p.a.75.12	yes	112	11.9	even	5	inner