Embedded newform 77.4.n.a.17.12

• Label: 77.4.n.a.17.12
• Level: $77$
• Weight: $4$
• Character: 77.17
• Analytic conductor: $4.543$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.54314707044\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(22\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			17.12
Character	\(\chi\)	\(=\)	77.17
Dual form			77.4.n.a.68.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.105220 + 0.0947403i) q^{2} +(2.36435 + 5.31043i) q^{3} +(-0.834132 - 7.93624i) q^{4} +(1.95794 + 9.21137i) q^{5} +(-0.254335 + 0.782762i) q^{6} +(12.6485 + 13.5283i) q^{7} +(1.32990 - 1.83045i) q^{8} +(-4.54395 + 5.04656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 5 q^{2} - 9 q^{3} - 93 q^{4} + 15 q^{5} - 100 q^{8} - 117 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.105220	+	0.0947403i	0.0372008	+	0.0334958i	0.687521	−	0.726164i	\(-0.258699\pi\)
							−0.650321	+	0.759660i	\(0.725365\pi\)
\(3\)	2.36435	+	5.31043i	0.455020	+	1.02199i	0.984774	+	0.173841i	\(0.0556178\pi\)
							−0.529753	+	0.848152i	\(0.677716\pi\)
\(5\)	1.95794	+	9.21137i	0.175123	+	0.823890i	0.974734	+	0.223371i	\(0.0717061\pi\)
							−0.799610	+	0.600519i	\(0.794961\pi\)
\(7\)	12.6485	+	13.5283i	0.682957	+	0.730458i
							\(11\)	−10.2088	+	35.0254i	−0.279826	+	0.960051i
\(13\)	8.20894	+	25.2645i	0.175135	+	0.539009i								0.999640	−	0.0268458i	\(-0.00854632\pi\)
							−0.824505	+	0.565855i	\(0.808546\pi\)
\(17\)	30.2014	+	33.5420i	0.430877	+	0.478538i	0.919011	−	0.394231i	\(-0.128989\pi\)
							−0.488134	+	0.872769i	\(0.662322\pi\)
\(19\)	13.0576	−	124.234i	0.157664	−	1.50007i	−0.574254	−	0.818677i	\(-0.694708\pi\)
							0.731918	−	0.681393i	\(-0.238625\pi\)
\(23\)	28.9275	+	50.1039i	0.262252	+	0.454234i	0.966840	−	0.255383i	\(-0.0822015\pi\)
							−0.704588	+	0.709617i	\(0.748868\pi\)
\(29\)	−163.455	−	224.976i	−1.04665	−	1.44059i	−0.891680	−	0.452667i	\(-0.850473\pi\)
							−0.154968	−	0.987920i	\(-0.549527\pi\)
\(31\)	45.7852	−	215.403i	0.265267	−	1.24798i	−0.620631	−	0.784103i	\(-0.713124\pi\)
							0.885898	−	0.463880i	\(-0.153543\pi\)
\(37\)	−180.798	−	80.4966i	−0.803326	−	0.357664i	−0.0363498	−	0.999339i	\(-0.511573\pi\)
							−0.766976	+	0.641675i	\(0.778240\pi\)
\(41\)	−203.686	−	147.987i	−0.775864	−	0.563698i	0.127871	−	0.991791i	\(-0.459186\pi\)
							−0.903735	+	0.428093i	\(0.859186\pi\)
\(43\)		−	206.507i		−	0.732373i	−0.930541	−	0.366187i	\(-0.880663\pi\)
							0.930541	−	0.366187i	\(-0.119337\pi\)
\(47\)	8.64670	+	0.908805i	0.0268351	+	0.00282049i	0.117937	−	0.993021i	\(-0.462372\pi\)
							−0.0911014	+	0.995842i	\(0.529039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.n.a.17.12	✓	176
7.5	odd	6	inner	77.4.n.a.61.12	yes	176
11.2	odd	10	inner	77.4.n.a.24.12	yes	176
77.68	even	30	inner	77.4.n.a.68.12	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.n.a.17.12	✓	176	1.1	even	1	trivial
77.4.n.a.24.12	yes	176	11.2	odd	10	inner
77.4.n.a.61.12	yes	176	7.5	odd	6	inner
77.4.n.a.68.12	yes	176	77.68	even	30	inner