Embedded newform 76.5.g.a.7.7

• Label: 76.5.g.a.7.7
• Level: $76$
• Weight: $5$
• Character: 76.7
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.54548 - 1.85191i) q^{2} +(-14.2858 + 8.24790i) q^{3} +(9.14084 + 13.1318i) q^{4} +(15.0446 + 26.0580i) q^{5} +(65.9243 - 2.78673i) q^{6} +53.2660i q^{7} +(-8.08964 - 63.4867i) q^{8} +(95.5557 - 165.507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.54548	−	1.85191i	−0.886370	−	0.462978i
							\(3\)	−14.2858	+	8.24790i	−1.58731	+	0.916433i	−0.593561	+	0.804789i	\(0.702278\pi\)
−0.993748	+	0.111644i	\(0.964388\pi\)
\(5\)	15.0446	+	26.0580i	0.601784	+	1.04232i	0.992551	+	0.121831i	\(0.0388767\pi\)
							−0.390766	+	0.920490i	\(0.627790\pi\)
\(7\)			53.2660i			1.08706i	0.839390	+	0.543530i	\(0.182913\pi\)
							−0.839390	+	0.543530i	\(0.817087\pi\)
\(11\)			194.395i			1.60657i	0.595594	+	0.803285i	\(0.296917\pi\)
							−0.595594	+	0.803285i	\(0.703083\pi\)
\(13\)	−106.699	+	184.807i	−0.631353	+	1.09353i	0.355923	+	0.934515i	\(0.384167\pi\)
							−0.987275	+	0.159020i	\(0.949167\pi\)
\(17\)	42.2299	+	73.1443i	0.146124	+	0.253094i	0.929792	−	0.368086i	\(-0.119987\pi\)
							−0.783668	+	0.621180i	\(0.786653\pi\)
\(19\)	83.7690	−	351.146i	0.232047	−	0.972705i
							\(23\)	−268.937	−	155.271i	−0.508388	−	0.293518i	0.223783	−	0.974639i	\(-0.428159\pi\)
−0.732171	+	0.681121i	\(0.761493\pi\)
\(29\)	−77.1824	+	133.684i	−0.0917745	+	0.158958i	−0.908258	−	0.418411i	\(-0.862587\pi\)
							0.816483	+	0.577369i	\(0.195921\pi\)
\(31\)		−	1187.59i		−	1.23579i	−0.786262	−	0.617894i	\(-0.787986\pi\)
							0.786262	−	0.617894i	\(-0.212014\pi\)
\(37\)	759.983			0.555137			0.277569	−	0.960706i	\(-0.410471\pi\)
							0.277569	+	0.960706i	\(0.410471\pi\)
\(41\)	627.053	+	1086.09i	0.373024	+	0.646096i	0.990029	−	0.140863i	\(-0.0449877\pi\)
							−0.617005	+	0.786959i	\(0.711654\pi\)
\(43\)	−479.506	+	276.843i	−0.259333	+	0.149726i	−0.624030	−	0.781400i	\(-0.714506\pi\)
							0.364697	+	0.931126i	\(0.381172\pi\)
\(47\)	713.038	+	411.673i	0.322788	+	0.186362i	0.652634	−	0.757673i	\(-0.273664\pi\)
							−0.329847	+	0.944034i	\(0.606997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.7	✓	76
4.3	odd	2	inner	76.5.g.a.7.21	yes	76
19.11	even	3	inner	76.5.g.a.11.21	yes	76
76.11	odd	6	inner	76.5.g.a.11.7	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.7	✓	76	1.1	even	1	trivial
76.5.g.a.7.21	yes	76	4.3	odd	2	inner
76.5.g.a.11.7	yes	76	76.11	odd	6	inner
76.5.g.a.11.21	yes	76	19.11	even	3	inner