Embedded newform 109.3.g.a.8.4

• Label: 109.3.g.a.8.4
• Level: $109$
• Weight: $3$
• Character: 109.8
• Analytic conductor: $2.970$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	109.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.97003488153\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.4
Character	\(\chi\)	\(=\)	109.8
Dual form			109.3.g.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98618 - 1.98618i) q^{2} +(2.72771 + 4.72454i) q^{3} +3.88979i q^{4} +(0.225365 + 0.390344i) q^{5} +(3.96604 - 14.8015i) q^{6} +(-0.538484 - 0.932682i) q^{7} +(-0.218902 + 0.218902i) q^{8} +(-10.3808 + 17.9801i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 2 q^{3} + 6 q^{5} - 4 q^{6} + 8 q^{7} + 6 q^{8} - 114 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98618	−	1.98618i	−0.993088	−	0.993088i	0.00688847	−	0.999976i	\(-0.497807\pi\)
							−0.999976	+	0.00688847i	\(0.997807\pi\)
\(3\)	2.72771	+	4.72454i	0.909237	+	1.57485i	0.815126	+	0.579284i	\(0.196668\pi\)
							0.0941114	+	0.995562i	\(0.469999\pi\)
\(5\)	0.225365	+	0.390344i	0.0450731	+	0.0780688i	0.887682	−	0.460457i	\(-0.152315\pi\)
							−0.842609	+	0.538526i	\(0.818981\pi\)
\(7\)	−0.538484	−	0.932682i	−0.0769263	−	0.133240i	0.824996	−	0.565139i	\(-0.191177\pi\)
							−0.901922	+	0.431898i	\(0.857844\pi\)
\(11\)	4.03055	+	15.0422i	0.366414	+	1.36747i	0.865494	+	0.500919i	\(0.167004\pi\)
							−0.499081	+	0.866556i	\(0.666329\pi\)
\(13\)	−1.21446	−	0.325413i	−0.0934199	−	0.0250318i	0.211806	−	0.977312i	\(-0.432065\pi\)
							−0.305226	+	0.952280i	\(0.598732\pi\)
\(17\)	−5.88130	−	5.88130i	−0.345959	−	0.345959i	0.512643	−	0.858602i	\(-0.328666\pi\)
							−0.858602	+	0.512643i	\(0.828666\pi\)
\(19\)	0.820717	−	0.820717i	0.0431956	−	0.0431956i	−0.685179	−	0.728375i	\(-0.740276\pi\)
							0.728375	+	0.685179i	\(0.240276\pi\)
\(23\)	28.9308	+	28.9308i	1.25786	+	1.25786i	0.952113	+	0.305746i	\(0.0989059\pi\)
							0.305746	+	0.952113i	\(0.401094\pi\)
\(29\)	−17.7165	+	10.2286i	−0.610914	+	0.352711i	−0.773323	−	0.634012i	\(-0.781407\pi\)
							0.162409	+	0.986724i	\(0.448074\pi\)
\(31\)	−0.274007	−	0.158198i	−0.00883895	−	0.00510317i	0.495574	−	0.868566i	\(-0.334958\pi\)
							−0.504413	+	0.863463i	\(0.668291\pi\)
\(37\)	39.7133	−	10.6411i	1.07333	−	0.287598i	0.321471	−	0.946920i	\(-0.395823\pi\)
							0.751862	+	0.659321i	\(0.229156\pi\)
\(41\)	−36.0502	−	36.0502i	−0.879274	−	0.879274i	0.114185	−	0.993459i	\(-0.463574\pi\)
							−0.993459	+	0.114185i	\(0.963574\pi\)
\(43\)		−	55.2934i		−	1.28589i	−0.765911	−	0.642947i	\(-0.777712\pi\)
							0.765911	−	0.642947i	\(-0.222288\pi\)
\(47\)	54.0837	−	14.4917i	1.15072	−	0.308334i	0.367464	−	0.930038i	\(-0.380226\pi\)
							0.783252	+	0.621704i	\(0.213559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.3.g.a.8.4	✓	72
109.41	odd	12	inner	109.3.g.a.41.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.3.g.a.8.4	✓	72	1.1	even	1	trivial
109.3.g.a.41.4	yes	72	109.41	odd	12	inner