Embedded newform 76.8.f.a.27.19

• Label: 76.8.f.a.27.19
• Level: $76$
• Weight: $8$
• Character: 76.27
• Analytic conductor: $23.741$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.7412619368\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.19
Character	\(\chi\)	\(=\)	76.27
Dual form			76.8.f.a.31.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.31847 + 8.62786i) q^{2} +(-23.5556 + 40.7996i) q^{3} +(-20.8799 - 126.286i) q^{4} +(-56.7551 + 98.3028i) q^{5} +(-179.622 - 501.825i) q^{6} +1227.46i q^{7} +(1242.38 + 744.068i) q^{8} +(-16.2362 - 28.1219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 3 q^{2} - 27 q^{4} - 2 q^{5} + 309 q^{6} - 45540 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.31847	+	8.62786i	−0.646868	+	0.762602i
							\(3\)	−23.5556	+	40.7996i	−0.503698	+	0.872431i	0.496293	+	0.868155i	\(0.334694\pi\)
−0.999991	+	0.00427571i	\(0.998639\pi\)
\(5\)	−56.7551	+	98.3028i	−0.203053	+	0.351699i	−0.949511	−	0.313735i	\(-0.898420\pi\)
							0.746457	+	0.665433i	\(0.231753\pi\)
\(7\)			1227.46i			1.35258i	0.736635	+	0.676291i	\(0.236414\pi\)
							−0.736635	+	0.676291i	\(0.763586\pi\)
\(11\)			5539.82i			1.25494i	0.778642	+	0.627468i	\(0.215909\pi\)
							−0.778642	+	0.627468i	\(0.784091\pi\)
\(13\)	−4837.20	+	2792.76i	−0.610651	+	0.352559i	−0.773220	−	0.634138i	\(-0.781355\pi\)
							0.162569	+	0.986697i	\(0.448022\pi\)
\(17\)	−12586.1	+	21799.7i	−0.621324	+	1.07617i	0.367915	+	0.929859i	\(0.380072\pi\)
							−0.989239	+	0.146306i	\(0.953262\pi\)
\(19\)	15003.6	+	25860.5i	0.501830	+	0.864966i
							\(23\)	23731.4	−	13701.3i	0.406701	−	0.234809i	−0.282670	−	0.959217i	\(-0.591220\pi\)
0.689371	+	0.724408i	\(0.257887\pi\)
\(29\)	130106.	−	75116.8i	0.990615	−	0.571932i	0.0851567	−	0.996368i	\(-0.472861\pi\)
							0.905458	+	0.424436i	\(0.139528\pi\)
\(31\)	139199.			0.839207			0.419604	−	0.907707i	\(-0.362169\pi\)
							0.419604	+	0.907707i	\(0.362169\pi\)
\(37\)		−	126301.i		−	0.409921i	−0.978770	−	0.204961i	\(-0.934293\pi\)
							0.978770	−	0.204961i	\(-0.0657066\pi\)
\(41\)	−85701.5	−	49479.8i	−0.194198	−	0.112120i	0.399748	−	0.916625i	\(-0.369098\pi\)
							−0.593946	+	0.804505i	\(0.702431\pi\)
\(43\)	40181.8	+	23199.0i	0.0770707	+	0.0444968i	0.538040	−	0.842919i	\(-0.319165\pi\)
							−0.460969	+	0.887416i	\(0.652498\pi\)
\(47\)	124418.	−	71833.0i	0.174800	−	0.100921i	−0.410047	−	0.912064i	\(-0.634488\pi\)
							0.584847	+	0.811143i	\(0.301154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.f.a.27.19	✓	136
4.3	odd	2	inner	76.8.f.a.27.41	yes	136
19.12	odd	6	inner	76.8.f.a.31.41	yes	136
76.31	even	6	inner	76.8.f.a.31.19	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.f.a.27.19	✓	136	1.1	even	1	trivial
76.8.f.a.27.41	yes	136	4.3	odd	2	inner
76.8.f.a.31.19	yes	136	76.31	even	6	inner
76.8.f.a.31.41	yes	136	19.12	odd	6	inner