Embedded newform 109.3.g.a.8.1

• Label: 109.3.g.a.8.1
• 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			8.1
Character	\(\chi\)	\(=\)	109.8
Dual form			109.3.g.a.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79587 - 2.79587i) q^{2} +(0.494454 + 0.856419i) q^{3} +11.6337i q^{4} +(-0.685437 - 1.18721i) q^{5} +(1.01201 - 3.77686i) q^{6} +(1.20222 + 2.08230i) q^{7} +(21.3429 - 21.3429i) q^{8} +(4.01103 - 6.94731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 2 q^{3} + 6 q^{5} - 4 q^{6} + 8 q^{7} + 6 q^{8} - 114 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79587	−	2.79587i	−1.39793	−	1.39793i	−0.805953	−	0.591980i	\(-0.798347\pi\)
							−0.591980	−	0.805953i	\(-0.701653\pi\)
\(3\)	0.494454	+	0.856419i	0.164818	+	0.285473i	0.936591	−	0.350425i	\(-0.113963\pi\)
							−0.771773	+	0.635899i	\(0.780630\pi\)
\(5\)	−0.685437	−	1.18721i	−0.137087	−	0.237442i	0.789306	−	0.614001i	\(-0.210441\pi\)
							−0.926393	+	0.376558i	\(0.877107\pi\)
\(7\)	1.20222	+	2.08230i	0.171745	+	0.297472i	0.939030	−	0.343835i	\(-0.111726\pi\)
							−0.767285	+	0.641307i	\(0.778393\pi\)
\(11\)	−2.02998	−	7.57600i	−0.184544	−	0.688727i	−0.994728	−	0.102551i	\(-0.967299\pi\)
							0.810184	−	0.586176i	\(-0.199367\pi\)
\(13\)	18.5980	+	4.98331i	1.43061	+	0.383332i	0.889236	−	0.457448i	\(-0.151236\pi\)
							0.541377	+	0.840780i	\(0.317903\pi\)
\(17\)	−12.8704	−	12.8704i	−0.757083	−	0.757083i	0.218707	−	0.975791i	\(-0.429816\pi\)
							−0.975791	+	0.218707i	\(0.929816\pi\)
\(19\)	7.84257	−	7.84257i	0.412767	−	0.412767i	−0.469934	−	0.882701i	\(-0.655722\pi\)
							0.882701	+	0.469934i	\(0.155722\pi\)
\(23\)	14.2848	+	14.2848i	0.621077	+	0.621077i	0.945807	−	0.324730i	\(-0.105273\pi\)
							−0.324730	+	0.945807i	\(0.605273\pi\)
\(29\)	30.2472	−	17.4632i	1.04301	−	0.602180i	0.122323	−	0.992490i	\(-0.460966\pi\)
							0.920683	+	0.390311i	\(0.127632\pi\)
\(31\)	−12.0995	−	6.98563i	−0.390305	−	0.225343i	0.291987	−	0.956422i	\(-0.405684\pi\)
							−0.682292	+	0.731079i	\(0.739017\pi\)
\(37\)	−14.2831	+	3.82714i	−0.386029	+	0.103436i	−0.446614	−	0.894727i	\(-0.647370\pi\)
							0.0605847	+	0.998163i	\(0.480703\pi\)
\(41\)	38.1492	+	38.1492i	0.930469	+	0.930469i	0.997735	−	0.0672662i	\(-0.0214277\pi\)
							−0.0672662	+	0.997735i	\(0.521428\pi\)
\(43\)		−	32.3132i		−	0.751470i	−0.926727	−	0.375735i	\(-0.877390\pi\)
							0.926727	−	0.375735i	\(-0.122610\pi\)
\(47\)	−60.8201	+	16.2967i	−1.29405	+	0.346738i	−0.839195	−	0.543830i	\(-0.816974\pi\)
							−0.454850	+	0.890568i	\(0.650307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.3.g.a.8.1	✓	72
109.41	odd	12	inner	109.3.g.a.41.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.3.g.a.8.1	✓	72	1.1	even	1	trivial
109.3.g.a.41.1	yes	72	109.41	odd	12	inner