Embedded newform 76.8.f.a.27.5

• Label: 76.8.f.a.27.5
• 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			27.5
Character	\(\chi\)	\(=\)	76.27
Dual form			76.8.f.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.9494 - 2.84804i) q^{2} +(35.9354 - 62.2419i) q^{3} +(111.777 + 62.3685i) q^{4} +(-104.293 + 180.641i) q^{5} +(-570.738 + 579.164i) q^{6} +565.242i q^{7} +(-1046.26 - 1001.24i) q^{8} +(-1489.21 - 2579.38i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 3 q^{2} - 27 q^{4} - 2 q^{5} + 309 q^{6} - 45540 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.9494	−	2.84804i	−0.967797	−	0.251734i
							\(3\)	35.9354	−	62.2419i	0.768419	−	1.33094i	−0.170001	−	0.985444i	\(-0.554377\pi\)
0.938420	−	0.345497i	\(-0.112290\pi\)
\(5\)	−104.293	+	180.641i	−0.373131	+	0.646282i	−0.990045	−	0.140748i	\(-0.955049\pi\)
							0.616914	+	0.787030i	\(0.288382\pi\)
\(7\)			565.242i			0.622862i	0.950269	+	0.311431i	\(0.100808\pi\)
							−0.950269	+	0.311431i	\(0.899192\pi\)
\(11\)			2649.53i			0.600198i	0.953908	+	0.300099i	\(0.0970197\pi\)
							−0.953908	+	0.300099i	\(0.902980\pi\)
\(13\)	10837.1	−	6256.81i	1.36808	−	0.789861i	0.377397	−	0.926051i	\(-0.376819\pi\)
							0.990683	+	0.136190i	\(0.0434857\pi\)
\(17\)	2081.59	−	3605.42i	0.102760	−	0.177986i	−0.810061	−	0.586346i	\(-0.800566\pi\)
							0.912821	+	0.408360i	\(0.133899\pi\)
\(19\)	−416.797	−	29894.8i	−0.0139408	−	0.999903i
							\(23\)	−13574.1	+	7837.01i	−0.232629	+	0.134308i	−0.611784	−	0.791025i	\(-0.709548\pi\)
0.379155	+	0.925333i	\(0.376215\pi\)
\(29\)	124164.	−	71686.3i	0.945375	−	0.545812i	0.0537336	−	0.998555i	\(-0.482888\pi\)
							0.891641	+	0.452743i	\(0.149554\pi\)
\(31\)	191562.			1.15490			0.577448	−	0.816428i	\(-0.304049\pi\)
							0.577448	+	0.816428i	\(0.304049\pi\)
\(37\)		−	557722.i		−	1.81014i	−0.425264	−	0.905070i	\(-0.639819\pi\)
							0.425264	−	0.905070i	\(-0.360181\pi\)
\(41\)	98387.5	+	56804.1i	0.222944	+	0.128717i	0.607313	−	0.794463i	\(-0.292247\pi\)
							−0.384368	+	0.923180i	\(0.625581\pi\)
\(43\)	−618718.	−	357217.i	−1.18673	−	0.685161i	−0.229171	−	0.973386i	\(-0.573602\pi\)
							−0.957563	+	0.288225i	\(0.906935\pi\)
\(47\)	247763.	−	143046.i	0.348092	−	0.200971i	−0.315753	−	0.948842i	\(-0.602257\pi\)
							0.663845	+	0.747871i	\(0.268924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.f.a.27.5	✓	136
4.3	odd	2	inner	76.8.f.a.27.18	yes	136
19.12	odd	6	inner	76.8.f.a.31.18	yes	136
76.31	even	6	inner	76.8.f.a.31.5	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.f.a.27.5	✓	136	1.1	even	1	trivial
76.8.f.a.27.18	yes	136	4.3	odd	2	inner
76.8.f.a.31.5	yes	136	76.31	even	6	inner
76.8.f.a.31.18	yes	136	19.12	odd	6	inner