Embedded newform 109.2.f.a.27.1

• Label: 109.2.f.a.27.1
• 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			27.1
Character	\(\chi\)	\(=\)	109.27
Dual form			109.2.f.a.105.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13355 - 1.96336i) q^{2} +(1.99711 + 0.726890i) q^{3} +(-1.56986 + 2.71908i) q^{4} +(0.184738 - 0.155013i) q^{5} +(-0.836675 - 4.74502i) q^{6} +(2.60110 - 2.18258i) q^{7} +2.58386 q^{8} +(1.16196 + 0.974999i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13355	−	1.96336i	−0.801539	−	1.38831i	−0.918603	−	0.395182i	\(-0.870681\pi\)
							0.117064	−	0.993124i	\(-0.462652\pi\)
\(3\)	1.99711	+	0.726890i	1.15303	+	0.419670i	0.846604	−	0.532223i	\(-0.178643\pi\)
							0.306430	+	0.951893i	\(0.400866\pi\)
\(5\)	0.184738	−	0.155013i	0.0826172	−	0.0693241i	−0.600544	−	0.799592i	\(-0.705049\pi\)
							0.683161	+	0.730268i	\(0.260605\pi\)
\(7\)	2.60110	−	2.18258i	0.983124	−	0.824939i	−0.00143392	−	0.999999i	\(-0.500456\pi\)
							0.984558	+	0.175060i	\(0.0560120\pi\)
\(11\)	0.418587	−	0.152353i	0.126209	−	0.0459362i	−0.278144	−	0.960539i	\(-0.589719\pi\)
							0.404352	+	0.914603i	\(0.367497\pi\)
\(13\)	−3.69828	+	3.10323i	−1.02572	+	0.860680i	−0.990335	−	0.138694i	\(-0.955710\pi\)
							−0.0353830	+	0.999374i	\(0.511265\pi\)
\(17\)	−3.08657	−	5.34610i	−0.748604	−	1.29662i	−0.948492	−	0.316801i	\(-0.897391\pi\)
							0.199888	−	0.979819i	\(-0.435942\pi\)
\(19\)	2.39000	+	4.13961i	0.548304	+	0.949691i	0.998391	+	0.0567059i	\(0.0180597\pi\)
							−0.450087	+	0.892985i	\(0.648607\pi\)
\(23\)	−2.10106	+	3.63915i	−0.438102	+	0.758815i	−0.997543	−	0.0700554i	\(-0.977682\pi\)
							0.559441	+	0.828870i	\(0.311016\pi\)
\(29\)	4.38134	+	1.59468i	0.813595	+	0.296124i	0.715108	−	0.699014i	\(-0.246377\pi\)
							0.0984870	+	0.995138i	\(0.468600\pi\)
\(31\)	−0.471360	−	0.395518i	−0.0846587	−	0.0710371i	0.599476	−	0.800393i	\(-0.295376\pi\)
							−0.684135	+	0.729356i	\(0.739820\pi\)
\(37\)	5.79949	+	4.86635i	0.953430	+	0.800022i	0.979872	−	0.199628i	\(-0.0639734\pi\)
							−0.0264422	+	0.999650i	\(0.508418\pi\)
\(41\)	−9.81983			−1.53360			−0.766800	−	0.641886i	\(-0.778152\pi\)
							−0.766800	+	0.641886i	\(0.778152\pi\)
\(43\)	−4.09031	−	7.08463i	−0.623767	−	1.08040i	−0.988778	−	0.149392i	\(-0.952268\pi\)
							0.365011	−	0.931003i	\(-0.381065\pi\)
\(47\)	−0.934336	+	5.29888i	−0.136287	+	0.772921i	0.837668	+	0.546180i	\(0.183918\pi\)
							−0.973955	+	0.226742i	\(0.927193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.27.1	✓	42
3.2	odd	2		981.2.w.a.136.7		42
109.105	even	9	inner	109.2.f.a.105.1	yes	42
327.323	odd	18		981.2.w.a.541.7		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.27.1	✓	42	1.1	even	1	trivial
109.2.f.a.105.1	yes	42	109.105	even	9	inner
981.2.w.a.136.7		42	3.2	odd	2
981.2.w.a.541.7		42	327.323	odd	18