Embedded newform 76.6.f.a.27.12

• Label: 76.6.f.a.27.12
• 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			27.12
Character	\(\chi\)	\(=\)	76.27
Dual form			76.6.f.a.31.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.29234 + 3.68454i) q^{2} +(-1.79179 + 3.10347i) q^{3} +(4.84840 - 31.6306i) q^{4} +(-10.1109 + 17.5126i) q^{5} +(-3.74388 - 19.9231i) q^{6} -149.904i q^{7} +(95.7329 + 153.633i) q^{8} +(115.079 + 199.323i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.29234	+	3.68454i	−0.758786	+	0.651340i
							\(3\)	−1.79179	+	3.10347i	−0.114943	+	0.199088i	−0.917757	−	0.397142i	\(-0.870002\pi\)
0.802814	+	0.596230i	\(0.203335\pi\)
\(5\)	−10.1109	+	17.5126i	−0.180870	+	0.313276i	−0.942177	−	0.335116i	\(-0.891225\pi\)
							0.761307	+	0.648391i	\(0.224558\pi\)
\(7\)		−	149.904i		−	1.15629i	−0.815933	−	0.578147i	\(-0.803776\pi\)
							0.815933	−	0.578147i	\(-0.196224\pi\)
\(11\)			123.651i			0.308117i	0.988062	+	0.154058i	\(0.0492344\pi\)
							−0.988062	+	0.154058i	\(0.950766\pi\)
\(13\)	412.543	−	238.182i	0.677035	−	0.390886i	−0.121702	−	0.992567i	\(-0.538835\pi\)
							0.798737	+	0.601681i	\(0.205502\pi\)
\(17\)	−809.296	+	1401.74i	−0.679180	+	1.17637i	0.296048	+	0.955173i	\(0.404331\pi\)
							−0.975228	+	0.221201i	\(0.929002\pi\)
\(19\)	1555.11	−	240.301i	0.988271	−	0.152712i
							\(23\)	−3750.75	+	2165.50i	−1.47842	+	0.853568i	−0.999702	−	0.0243993i	\(-0.992233\pi\)
−0.478721	+	0.877967i	\(0.658899\pi\)
\(29\)	−6567.24	+	3791.60i	−1.45007	+	0.837196i	−0.998484	−	0.0550339i	\(-0.982473\pi\)
							−0.451581	+	0.892230i	\(0.649140\pi\)
\(31\)	−3220.24			−0.601844			−0.300922	−	0.953649i	\(-0.597294\pi\)
							−0.300922	+	0.953649i	\(0.597294\pi\)
\(37\)			5803.44i			0.696918i	0.937324	+	0.348459i	\(0.113295\pi\)
							−0.937324	+	0.348459i	\(0.886705\pi\)
\(41\)	15899.2	+	9179.42i	1.47712	+	0.852816i	0.999666	−	0.0258403i	\(-0.00822615\pi\)
							0.477455	+	0.878656i	\(0.341559\pi\)
\(43\)	9282.28	+	5359.13i	0.765568	+	0.442001i	0.831291	−	0.555837i	\(-0.187602\pi\)
							−0.0657235	+	0.997838i	\(0.520936\pi\)
\(47\)	25805.1	−	14898.6i	1.70396	−	0.983784i	0.762304	−	0.647219i	\(-0.224068\pi\)
							0.941660	−	0.336565i	\(-0.109265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.27.12	✓	96
4.3	odd	2	inner	76.6.f.a.27.27	yes	96
19.12	odd	6	inner	76.6.f.a.31.27	yes	96
76.31	even	6	inner	76.6.f.a.31.12	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.12	✓	96	1.1	even	1	trivial
76.6.f.a.27.27	yes	96	4.3	odd	2	inner
76.6.f.a.31.12	yes	96	76.31	even	6	inner
76.6.f.a.31.27	yes	96	19.12	odd	6	inner