Embedded newform 76.6.f.a.31.14

• Label: 76.6.f.a.31.14
• Level: $76$
• Weight: $6$
• Character: 76.31
• 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			31.14
Character	\(\chi\)	\(=\)	76.31
Dual form			76.6.f.a.27.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.14758 + 4.70029i) q^{2} +(5.80366 + 10.0522i) q^{3} +(-12.1854 - 29.5891i) q^{4} +(-48.4181 - 83.8626i) q^{5} +(-65.5159 - 4.36135i) q^{6} +77.0285i q^{7} +(177.432 + 35.8590i) q^{8} +(54.1352 - 93.7648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.14758	+	4.70029i	−0.556419	+	0.830902i
							\(3\)	5.80366	+	10.0522i	0.372305	+	0.644851i	0.989920	−	0.141630i	\(-0.0452343\pi\)
−0.617615	+	0.786481i	\(0.711901\pi\)
\(5\)	−48.4181	−	83.8626i	−0.866130	−	1.50018i	−0.865921	−	0.500180i	\(-0.833267\pi\)
							−0.000208377	−	1.00000i	\(-0.500066\pi\)
\(7\)			77.0285i			0.594164i	0.954852	+	0.297082i	\(0.0960135\pi\)
							−0.954852	+	0.297082i	\(0.903986\pi\)
\(11\)			612.215i			1.52553i	0.646673	+	0.762767i	\(0.276160\pi\)
							−0.646673	+	0.762767i	\(0.723840\pi\)
\(13\)	784.662	+	453.025i	1.28773	+	0.743470i	0.978249	−	0.207436i	\(-0.0665119\pi\)
							0.309479	+	0.950906i	\(0.399845\pi\)
\(17\)	−469.295	−	812.843i	−0.393844	−	0.682157i	0.599109	−	0.800667i	\(-0.295522\pi\)
							−0.992953	+	0.118510i	\(0.962188\pi\)
\(19\)	1573.54	−	8.46536i	0.999986	−	0.00537974i
							\(23\)	156.096	+	90.1218i	0.0615277	+	0.0355231i	0.530448	−	0.847717i	\(-0.322024\pi\)
−0.468921	+	0.883240i	\(0.655357\pi\)
\(29\)	5138.90	+	2966.95i	1.13469	+	0.655111i	0.945109	−	0.326755i	\(-0.105955\pi\)
							0.189576	+	0.981866i	\(0.439289\pi\)
\(31\)	4142.34			0.774179			0.387089	−	0.922042i	\(-0.373480\pi\)
							0.387089	+	0.922042i	\(0.373480\pi\)
\(37\)		−	573.009i		−	0.0688109i	−0.999408	−	0.0344055i	\(-0.989046\pi\)
							0.999408	−	0.0344055i	\(-0.0109538\pi\)
\(41\)	2742.72	−	1583.51i	0.254814	−	0.147117i	−0.367153	−	0.930161i	\(-0.619667\pi\)
							0.621966	+	0.783044i	\(0.286334\pi\)
\(43\)	−11790.0	+	6806.98i	−0.972398	+	0.561414i	−0.899966	−	0.435959i	\(-0.856409\pi\)
							−0.0724313	+	0.997373i	\(0.523076\pi\)
\(47\)	−1774.06	−	1024.25i	−0.117145	−	0.0676335i	0.440283	−	0.897859i	\(-0.354878\pi\)
							−0.557427	+	0.830226i	\(0.688212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.31.14	yes	96
4.3	odd	2	inner	76.6.f.a.31.3	yes	96
19.8	odd	6	inner	76.6.f.a.27.3	✓	96
76.27	even	6	inner	76.6.f.a.27.14	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.3	✓	96	19.8	odd	6	inner
76.6.f.a.27.14	yes	96	76.27	even	6	inner
76.6.f.a.31.3	yes	96	4.3	odd	2	inner
76.6.f.a.31.14	yes	96	1.1	even	1	trivial