Embedded newform 77.5.g.a.12.17

• Label: 77.5.g.a.12.17
• Level: $77$
• Weight: $5$
• Character: 77.12
• Analytic conductor: $7.959$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	77.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.95948715746\)
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			12.17
Character	\(\chi\)	\(=\)	77.12
Dual form			77.5.g.a.45.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21125 - 2.09794i) q^{2} +(0.957269 - 0.552680i) q^{3} +(5.06576 + 8.77415i) q^{4} +(15.8155 + 9.13109i) q^{5} -2.67773i q^{6} +(-36.5703 + 32.6131i) q^{7} +63.3035 q^{8} +(-39.8891 + 69.0899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 18 q^{3} - 192 q^{4} - 54 q^{7} - 180 q^{8} + 680 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21125	−	2.09794i	0.302812	−	0.524486i	−0.673960	−	0.738768i	\(-0.735408\pi\)
							0.976772	+	0.214282i	\(0.0687413\pi\)
\(3\)	0.957269	−	0.552680i	0.106363	−	0.0614088i	−0.445875	−	0.895095i	\(-0.647107\pi\)
							0.552238	+	0.833687i	\(0.313774\pi\)
\(5\)	15.8155	+	9.13109i	0.632621	+	0.365244i	0.781766	−	0.623571i	\(-0.214319\pi\)
							−0.149146	+	0.988815i	\(0.547652\pi\)
\(7\)	−36.5703	+	32.6131i	−0.746333	+	0.665572i
							\(11\)	18.2414	+	31.5951i	0.150756	+	0.261116i
\(13\)			40.0528i			0.236999i								0.992954	+	0.118500i	\(0.0378084\pi\)
							−0.992954	+	0.118500i	\(0.962192\pi\)
\(17\)	357.204	−	206.232i	1.23600	−	0.713605i	0.267726	−	0.963495i	\(-0.413728\pi\)
							0.968274	+	0.249890i	\(0.0803946\pi\)
\(19\)	199.395	+	115.121i	0.552341	+	0.318894i	0.750066	−	0.661364i	\(-0.230022\pi\)
							−0.197725	+	0.980258i	\(0.563355\pi\)
\(23\)	305.839	−	529.728i	0.578145	−	1.00138i	−0.417547	−	0.908655i	\(-0.637110\pi\)
							0.995692	−	0.0927210i	\(-0.0295564\pi\)
\(29\)	3.36263			0.00399837			0.00199919	−	0.999998i	\(-0.499364\pi\)
							0.00199919	+	0.999998i	\(0.499364\pi\)
\(31\)	−870.963	+	502.851i	−0.906309	+	0.523258i	−0.879242	−	0.476376i	\(-0.841950\pi\)
							−0.0270673	+	0.999634i	\(0.508617\pi\)
\(37\)	420.170	−	727.755i	0.306917	−	0.531596i	−0.670769	−	0.741666i	\(-0.734036\pi\)
							0.977686	+	0.210070i	\(0.0673692\pi\)
\(41\)			349.472i			0.207895i	0.994583	+	0.103948i	\(0.0331474\pi\)
							−0.994583	+	0.103948i	\(0.966853\pi\)
\(43\)	−1104.21			−0.597191			−0.298595	−	0.954380i	\(-0.596518\pi\)
							−0.298595	+	0.954380i	\(0.596518\pi\)
\(47\)	−809.140	−	467.157i	−0.366292	−	0.211479i	0.305545	−	0.952178i	\(-0.401161\pi\)
							−0.671838	+	0.740699i	\(0.734495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.5.g.a.12.17	✓	52
7.3	odd	6	inner	77.5.g.a.45.17	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.5.g.a.12.17	✓	52	1.1	even	1	trivial
77.5.g.a.45.17	yes	52	7.3	odd	6	inner