Embedded newform 76.6.f.a.27.9

• Label: 76.6.f.a.27.9
• Level: $76$
• Weight: $6$
• Character: 76.27
• Analytic conductor: $12.189$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1891703058\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.9
Character	\(\chi\)	\(=\)	76.27
Dual form			76.6.f.a.31.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.74667 - 3.07720i) q^{2} +(-3.22504 + 5.58593i) q^{3} +(13.0617 + 29.2129i) q^{4} +(-2.36309 + 4.09299i) q^{5} +(32.4972 - 16.5905i) q^{6} +80.2152i q^{7} +(27.8940 - 178.857i) q^{8} +(100.698 + 174.414i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.74667	−	3.07720i	−0.839100	−	0.543977i
							\(3\)	−3.22504	+	5.58593i	−0.206886	+	0.358338i	−0.950732	−	0.310014i	\(-0.899666\pi\)
0.743846	+	0.668351i	\(0.233000\pi\)
\(5\)	−2.36309	+	4.09299i	−0.0422722	+	0.0732177i	−0.886387	−	0.462944i	\(-0.846793\pi\)
							0.844115	+	0.536162i	\(0.180126\pi\)
\(7\)			80.2152i			0.618745i	0.950941	+	0.309372i	\(0.100119\pi\)
							−0.950941	+	0.309372i	\(0.899881\pi\)
\(11\)		−	518.517i		−	1.29206i	−0.763313	−	0.646028i	\(-0.776429\pi\)
							0.763313	−	0.646028i	\(-0.223571\pi\)
\(13\)	−282.141	+	162.894i	−0.463028	+	0.267329i	−0.713317	−	0.700842i	\(-0.752808\pi\)
							0.250289	+	0.968171i	\(0.419474\pi\)
\(17\)	386.161	−	668.851i	0.324076	−	0.561315i	−0.657249	−	0.753673i	\(-0.728280\pi\)
							0.981325	+	0.192358i	\(0.0616135\pi\)
\(19\)	−1558.84	+	214.749i	−0.990644	+	0.136473i
							\(23\)	−3886.24	+	2243.72i	−1.53183	+	0.884402i	−0.532552	+	0.846397i	\(0.678767\pi\)
−0.999278	+	0.0380048i	\(0.987900\pi\)
\(29\)	−2657.17	+	1534.12i	−0.586712	+	0.338738i	−0.763796	−	0.645457i	\(-0.776667\pi\)
							0.177084	+	0.984196i	\(0.443333\pi\)
\(31\)	−5934.95			−1.10921			−0.554604	−	0.832114i	\(-0.687130\pi\)
							−0.554604	+	0.832114i	\(0.687130\pi\)
\(37\)		−	9616.57i		−	1.15482i	−0.816453	−	0.577412i	\(-0.804063\pi\)
							0.816453	−	0.577412i	\(-0.195937\pi\)
\(41\)	−3908.01	−	2256.29i	−0.363075	−	0.209621i	0.307354	−	0.951595i	\(-0.400556\pi\)
							−0.670429	+	0.741974i	\(0.733890\pi\)
\(43\)	−17637.0	−	10182.7i	−1.45463	−	0.839832i	−0.455893	−	0.890035i	\(-0.650680\pi\)
							−0.998739	+	0.0502024i	\(0.984013\pi\)
\(47\)	7408.13	−	4277.09i	0.489175	−	0.282425i	−0.235057	−	0.971982i	\(-0.575528\pi\)
							0.724232	+	0.689556i	\(0.242194\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.27.9	yes	96
4.3	odd	2	inner	76.6.f.a.27.7	✓	96
19.12	odd	6	inner	76.6.f.a.31.7	yes	96
76.31	even	6	inner	76.6.f.a.31.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.7	✓	96	4.3	odd	2	inner
76.6.f.a.27.9	yes	96	1.1	even	1	trivial
76.6.f.a.31.7	yes	96	19.12	odd	6	inner
76.6.f.a.31.9	yes	96	76.31	even	6	inner