Embedded newform 77.3.p.a.3.9

• Label: 77.3.p.a.3.9
• Level: $77$
• Weight: $3$
• Character: 77.3
• 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			3.9
Character	\(\chi\)	\(=\)	77.3
Dual form			77.3.p.a.26.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.710777 + 0.316458i) q^{2} +(0.455331 - 2.14216i) q^{3} +(-2.27146 - 2.52272i) q^{4} +(1.67507 + 0.176057i) q^{5} +(1.00154 - 1.37851i) q^{6} +(1.90441 - 6.73596i) q^{7} +(-1.77788 - 5.47176i) q^{8} +(3.84038 + 1.70985i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.710777	+	0.316458i	0.355388	+	0.158229i	0.576662	−	0.816983i	\(-0.304355\pi\)
							−0.221274	+	0.975212i	\(0.571021\pi\)
\(3\)	0.455331	−	2.14216i	0.151777	−	0.714054i	−0.834773	−	0.550594i	\(-0.814401\pi\)
							0.986550	−	0.163460i	\(-0.0522655\pi\)
\(5\)	1.67507	+	0.176057i	0.335015	+	0.0352115i	0.270543	−	0.962708i	\(-0.412797\pi\)
							0.0644722	+	0.997920i	\(0.479464\pi\)
\(7\)	1.90441	−	6.73596i	0.272059	−	0.962281i
							\(11\)	1.65274	+	10.8751i	0.150249	+	0.988648i
\(13\)	−8.10329	−	11.1532i	−0.623330	−	0.857940i								0.374260	−	0.927324i	\(-0.377897\pi\)
							−0.997590	+	0.0693840i	\(0.977897\pi\)
\(17\)	8.40628	+	18.8808i	0.494487	+	1.11064i	0.972631	+	0.232355i	\(0.0746431\pi\)
							−0.478144	+	0.878281i	\(0.658690\pi\)
\(19\)	21.0548	+	18.9579i	1.10815	+	0.997782i	0.999993	+	0.00364799i	\(0.00116119\pi\)
							0.108156	+	0.994134i	\(0.465505\pi\)
\(23\)	−7.68013	−	13.3024i	−0.333919	−	0.578364i	0.649358	−	0.760483i	\(-0.275038\pi\)
							−0.983277	+	0.182119i	\(0.941704\pi\)
\(29\)	−11.9196	+	36.6847i	−0.411020	+	1.26499i	0.504743	+	0.863270i	\(0.331587\pi\)
							−0.915763	+	0.401719i	\(0.868413\pi\)
\(31\)	22.0479	−	2.31732i	0.711221	−	0.0747524i	0.257993	−	0.966147i	\(-0.416939\pi\)
							0.453228	+	0.891394i	\(0.350272\pi\)
\(37\)	−32.9233	+	6.99807i	−0.889819	+	0.189137i	−0.630069	−	0.776539i	\(-0.716974\pi\)
							−0.259750	+	0.965676i	\(0.583640\pi\)
\(41\)	59.3477	−	19.2832i	1.44751	−	0.470323i	0.523277	−	0.852163i	\(-0.324709\pi\)
							0.924229	+	0.381840i	\(0.124709\pi\)
\(43\)	28.0088			0.651368			0.325684	−	0.945479i	\(-0.394405\pi\)
							0.325684	+	0.945479i	\(0.394405\pi\)
\(47\)	−5.83129	−	5.25052i	−0.124070	−	0.111713i	0.604760	−	0.796408i	\(-0.293269\pi\)
							−0.728830	+	0.684694i	\(0.759936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.3.p.a.3.9	✓	112
7.5	odd	6	inner	77.3.p.a.47.6	yes	112
11.4	even	5	inner	77.3.p.a.59.6	yes	112
77.26	odd	30	inner	77.3.p.a.26.9	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.3.p.a.3.9	✓	112	1.1	even	1	trivial
77.3.p.a.26.9	yes	112	77.26	odd	30	inner
77.3.p.a.47.6	yes	112	7.5	odd	6	inner
77.3.p.a.59.6	yes	112	11.4	even	5	inner