Embedded newform 76.6.f.a.27.3

• Label: 76.6.f.a.27.3
• 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			27.3
Character	\(\chi\)	\(=\)	76.27
Dual form			76.6.f.a.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.64436 - 0.375742i) q^{2} +(-5.80366 + 10.0522i) q^{3} +(31.7176 + 4.24165i) q^{4} +(-48.4181 + 83.8626i) q^{5} +(36.5350 - 54.5577i) q^{6} +77.0285i q^{7} +(-177.432 - 35.8590i) q^{8} +(54.1352 + 93.7648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.64436	−	0.375742i	−0.997792	−	0.0664224i
							\(3\)	−5.80366	+	10.0522i	−0.372305	+	0.644851i	−0.989920	−	0.141630i	\(-0.954766\pi\)
0.617615	+	0.786481i	\(0.288099\pi\)
\(5\)	−48.4181	+	83.8626i	−0.866130	+	1.50018i	−0.000208377		1.00000i	\(0.500066\pi\)
							−0.865921	+	0.500180i	\(0.833267\pi\)
\(7\)			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	0.309479	−	0.950906i	\(-0.399845\pi\)
							0.978249	+	0.207436i	\(0.0665119\pi\)
\(17\)	−469.295	+	812.843i	−0.393844	+	0.682157i	−0.992953	−	0.118510i	\(-0.962188\pi\)
							0.599109	+	0.800667i	\(0.295522\pi\)
\(19\)	−1573.54	−	8.46536i	−0.999986	−	0.00537974i
							\(23\)	−156.096	+	90.1218i	−0.0615277	+	0.0355231i	−0.530448	−	0.847717i	\(-0.677976\pi\)
0.468921	+	0.883240i	\(0.344643\pi\)
\(29\)	5138.90	−	2966.95i	1.13469	−	0.655111i	0.189576	−	0.981866i	\(-0.439289\pi\)
							0.945109	+	0.326755i	\(0.105955\pi\)
\(31\)	−4142.34			−0.774179			−0.387089	−	0.922042i	\(-0.626520\pi\)
							−0.387089	+	0.922042i	\(0.626520\pi\)
\(37\)			573.009i			0.0688109i	0.999408	+	0.0344055i	\(0.0109538\pi\)
							−0.999408	+	0.0344055i	\(0.989046\pi\)
\(41\)	2742.72	+	1583.51i	0.254814	+	0.147117i	0.621966	−	0.783044i	\(-0.286334\pi\)
							−0.367153	+	0.930161i	\(0.619667\pi\)
\(43\)	11790.0	+	6806.98i	0.972398	+	0.561414i	0.899966	−	0.435959i	\(-0.143591\pi\)
							0.0724313	+	0.997373i	\(0.476924\pi\)
\(47\)	1774.06	−	1024.25i	0.117145	−	0.0676335i	−0.440283	−	0.897859i	\(-0.645122\pi\)
							0.557427	+	0.830226i	\(0.311788\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.3	✓	96	1.1	even	1	trivial
76.6.f.a.27.14	yes	96	4.3	odd	2	inner
76.6.f.a.31.3	yes	96	76.31	even	6	inner
76.6.f.a.31.14	yes	96	19.12	odd	6	inner