Embedded newform 109.2.f.a.27.2

• Label: 109.2.f.a.27.2
• Level: $109$
• Weight: $2$
• Character: 109.27
• Analytic conductor: $0.870$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	109.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.870369382032\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			27.2
Character	\(\chi\)	\(=\)	109.27
Dual form			109.2.f.a.105.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.754385 - 1.30663i) q^{2} +(-0.821044 - 0.298836i) q^{3} +(-0.138193 + 0.239358i) q^{4} +(1.83302 - 1.53808i) q^{5} +(0.228915 + 1.29824i) q^{6} +(-0.324966 + 0.272679i) q^{7} -2.60054 q^{8} +(-1.71332 - 1.43765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.754385	−	1.30663i	−0.533431	−	0.923929i	−0.999238	−	0.0390427i	\(-0.987569\pi\)
							0.465807	−	0.884886i	\(-0.345764\pi\)
\(3\)	−0.821044	−	0.298836i	−0.474030	−	0.172533i	0.0939470	−	0.995577i	\(-0.470052\pi\)
							−0.567977	+	0.823044i	\(0.692274\pi\)
\(5\)	1.83302	−	1.53808i	0.819750	−	0.687852i	−0.133164	−	0.991094i	\(-0.542514\pi\)
							0.952913	+	0.303242i	\(0.0980692\pi\)
\(7\)	−0.324966	+	0.272679i	−0.122826	+	0.103063i	−0.702131	−	0.712047i	\(-0.747768\pi\)
							0.579306	+	0.815110i	\(0.303324\pi\)
\(11\)	−2.57735	+	0.938079i	−0.777101	+	0.282842i	−0.699963	−	0.714179i	\(-0.746800\pi\)
							−0.0771375	+	0.997020i	\(0.524578\pi\)
\(13\)	4.74975	−	3.98551i	1.31734	−	1.10538i	0.330482	−	0.943812i	\(-0.392789\pi\)
							0.986861	−	0.161570i	\(-0.0516556\pi\)
\(17\)	0.603120	+	1.04463i	0.146278	+	0.253361i	0.929849	−	0.367941i	\(-0.119937\pi\)
							−0.783571	+	0.621302i	\(0.786604\pi\)
\(19\)	3.68968	+	6.39071i	0.846470	+	1.46613i	0.884339	+	0.466846i	\(0.154610\pi\)
							−0.0378689	+	0.999283i	\(0.512057\pi\)
\(23\)	4.11885	−	7.13406i	0.858840	−	1.48755i	−0.0141960	−	0.999899i	\(-0.504519\pi\)
							0.873036	−	0.487656i	\(-0.162148\pi\)
\(29\)	1.32107	+	0.480829i	0.245316	+	0.0892877i	0.461752	−	0.887009i	\(-0.347221\pi\)
							−0.216436	+	0.976297i	\(0.569443\pi\)
\(31\)	2.08169	+	1.74674i	0.373882	+	0.313724i	0.810295	−	0.586022i	\(-0.199307\pi\)
							−0.436413	+	0.899746i	\(0.643751\pi\)
\(37\)	−4.34545	−	3.64627i	−0.714388	−	0.599442i	0.211439	−	0.977391i	\(-0.432185\pi\)
							−0.925826	+	0.377949i	\(0.876630\pi\)
\(41\)	3.77946			0.590253			0.295126	−	0.955458i	\(-0.404638\pi\)
							0.295126	+	0.955458i	\(0.404638\pi\)
\(43\)	2.53065	+	4.38322i	0.385921	+	0.668435i	0.991897	−	0.127048i	\(-0.0405503\pi\)
							−0.605975	+	0.795483i	\(0.707217\pi\)
\(47\)	0.393938	−	2.23413i	0.0574617	−	0.325881i	−0.942504	−	0.334196i	\(-0.891535\pi\)
							0.999965	+	0.00831413i	\(0.00264650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.27.2	✓	42
3.2	odd	2		981.2.w.a.136.6		42
109.105	even	9	inner	109.2.f.a.105.2	yes	42
327.323	odd	18		981.2.w.a.541.6		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.27.2	✓	42	1.1	even	1	trivial
109.2.f.a.105.2	yes	42	109.105	even	9	inner
981.2.w.a.136.6		42	3.2	odd	2
981.2.w.a.541.6		42	327.323	odd	18