Embedded newform 76.6.f.a.31.12

• Label: 76.6.f.a.31.12
• 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			31.12
Character	\(\chi\)	\(=\)	76.31
Dual form			76.6.f.a.27.12

$q$-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 \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.12	✓	96	76.27	even	6	inner
76.6.f.a.27.27	yes	96	19.8	odd	6	inner
76.6.f.a.31.12	yes	96	1.1	even	1	trivial
76.6.f.a.31.27	yes	96	4.3	odd	2	inner