Embedded newform 76.6.k.a.3.19

• Label: 76.6.k.a.3.19
• Level: $76$
• Weight: $6$
• Character: 76.3
• Analytic conductor: $12.189$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1891703058\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(48\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.19
Character	\(\chi\)	\(=\)	76.3
Dual form			76.6.k.a.51.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06442 + 5.26670i) q^{2} +(-1.18382 + 0.430875i) q^{3} +(-23.4763 - 21.7454i) q^{4} +(16.6949 - 94.6815i) q^{5} +(0.174611 - 7.12434i) q^{6} +(-127.080 + 73.3697i) q^{7} +(162.992 - 78.7512i) q^{8} +(-184.933 + 155.177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 6 q^{2} + 24 q^{4} - 12 q^{5} + 264 q^{6} - 9 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06442	+	5.26670i	−0.364942	+	0.931030i
							\(3\)	−1.18382	+	0.430875i	−0.0759421	+	0.0276407i	−0.379712	−	0.925105i	\(-0.623977\pi\)
0.303770	+	0.952745i	\(0.401755\pi\)
\(5\)	16.6949	−	94.6815i	0.298647	−	1.69371i	−0.353349	−	0.935492i	\(-0.614957\pi\)
							0.651996	−	0.758222i	\(-0.273932\pi\)
\(7\)	−127.080	+	73.3697i	−0.980239	+	0.565941i	−0.902342	−	0.431020i	\(-0.858154\pi\)
							−0.0778969	+	0.996961i	\(0.524821\pi\)
\(11\)	595.427	+	343.770i	1.48370	+	0.856616i	0.999828	−	0.0185224i	\(-0.00589620\pi\)
							0.483873	+	0.875138i	\(0.339230\pi\)
\(13\)	−142.728	+	392.141i	−0.234234	+	0.643553i	0.765766	+	0.643120i	\(0.222360\pi\)
							−1.00000		0.000433476i	\(0.999862\pi\)
\(17\)	721.140	+	605.108i	0.605198	+	0.507821i	0.893112	−	0.449835i	\(-0.148517\pi\)
							−0.287914	+	0.957656i	\(0.592962\pi\)
\(19\)	−1162.81	+	1060.18i	−0.738966	+	0.673743i
							\(23\)	3385.83	−	597.014i	1.33458	−	0.235323i	0.539584	−	0.841932i	\(-0.318582\pi\)
0.795000	+	0.606609i	\(0.207471\pi\)
\(29\)	−472.997	−	563.696i	−0.104439	−	0.124466i	0.711293	−	0.702896i	\(-0.248110\pi\)
							−0.815732	+	0.578430i	\(0.803666\pi\)
\(31\)	3050.82	+	5284.17i	0.570180	+	0.987581i	0.996547	+	0.0830306i	\(0.0264599\pi\)
							−0.426367	+	0.904550i	\(0.640207\pi\)
\(37\)			12169.0i			1.46134i	0.682732	+	0.730669i	\(0.260792\pi\)
							−0.682732	+	0.730669i	\(0.739208\pi\)
\(41\)	−4285.98	−	11775.6i	−0.398190	−	1.09402i	−0.963165	−	0.268911i	\(-0.913336\pi\)
							0.564975	−	0.825108i	\(-0.308886\pi\)
\(43\)	−3694.83	−	651.498i	−0.304736	−	0.0537331i	0.0191894	−	0.999816i	\(-0.493891\pi\)
							−0.323925	+	0.946083i	\(0.605003\pi\)
\(47\)	6182.92	+	7368.52i	0.408271	+	0.486559i	0.930523	−	0.366232i	\(-0.119352\pi\)
							−0.522252	+	0.852791i	\(0.674908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.k.a.3.19	yes	288
4.3	odd	2	inner	76.6.k.a.3.9	✓	288
19.13	odd	18	inner	76.6.k.a.51.9	yes	288
76.51	even	18	inner	76.6.k.a.51.19	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.k.a.3.9	✓	288	4.3	odd	2	inner
76.6.k.a.3.19	yes	288	1.1	even	1	trivial
76.6.k.a.51.9	yes	288	19.13	odd	18	inner
76.6.k.a.51.19	yes	288	76.51	even	18	inner