Embedded newform 109.2.f.a.27.4

• Label: 109.2.f.a.27.4
• 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			27.4
Character	\(\chi\)	\(=\)	109.27
Dual form			109.2.f.a.105.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.304565 - 0.527522i) q^{2} +(1.79458 + 0.653175i) q^{3} +(0.814480 - 1.41072i) q^{4} +(0.460326 - 0.386260i) q^{5} +(-0.202003 - 1.14562i) q^{6} +(-2.95565 + 2.48009i) q^{7} -2.21051 q^{8} +(0.495757 + 0.415990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.304565	−	0.527522i	−0.215360	−	0.373014i	0.738024	−	0.674775i	\(-0.235759\pi\)
							−0.953384	+	0.301760i	\(0.902426\pi\)
\(3\)	1.79458	+	0.653175i	1.03610	+	0.377111i	0.803402	−	0.595438i	\(-0.203021\pi\)
							0.232701	+	0.972548i	\(0.425244\pi\)
\(5\)	0.460326	−	0.386260i	0.205864	−	0.172741i	−0.534026	−	0.845468i	\(-0.679322\pi\)
							0.739891	+	0.672727i	\(0.234877\pi\)
\(7\)	−2.95565	+	2.48009i	−1.11713	+	0.937384i	−0.998456	−	0.0555491i	\(-0.982309\pi\)
							−0.118675	+	0.992933i	\(0.537865\pi\)
\(11\)	3.72438	−	1.35556i	1.12294	−	0.408718i	0.287217	−	0.957866i	\(-0.407270\pi\)
							0.835725	+	0.549148i	\(0.185048\pi\)
\(13\)	−2.58944	+	2.17279i	−0.718180	+	0.602625i	−0.926881	−	0.375355i	\(-0.877521\pi\)
							0.208701	+	0.977979i	\(0.433076\pi\)
\(17\)	2.43790	+	4.22257i	0.591278	+	1.02412i	0.994061	+	0.108827i	\(0.0347096\pi\)
							−0.402783	+	0.915295i	\(0.631957\pi\)
\(19\)	−2.93133	−	5.07722i	−0.672494	−	1.16479i	−0.977195	−	0.212345i	\(-0.931890\pi\)
							0.304701	−	0.952448i	\(-0.401444\pi\)
\(23\)	−1.40127	+	2.42707i	−0.292184	+	0.506078i	−0.974326	−	0.225142i	\(-0.927715\pi\)
							0.682142	+	0.731220i	\(0.261049\pi\)
\(29\)	1.37349	+	0.499909i	0.255051	+	0.0928308i	0.466381	−	0.884584i	\(-0.345557\pi\)
							−0.211331	+	0.977415i	\(0.567780\pi\)
\(31\)	1.06777	+	0.895964i	0.191777	+	0.160920i	0.733620	−	0.679560i	\(-0.237829\pi\)
							−0.541844	+	0.840479i	\(0.682274\pi\)
\(37\)	−6.69734	−	5.61974i	−1.10104	−	0.923879i	−0.103542	−	0.994625i	\(-0.533017\pi\)
							−0.997494	+	0.0707463i	\(0.977462\pi\)
\(41\)	4.49425			0.701884			0.350942	−	0.936397i	\(-0.385861\pi\)
							0.350942	+	0.936397i	\(0.385861\pi\)
\(43\)	1.79547	+	3.10984i	0.273806	+	0.474246i	0.969833	−	0.243769i	\(-0.0783840\pi\)
							−0.696027	+	0.718016i	\(0.745051\pi\)
\(47\)	1.93210	−	10.9575i	0.281825	−	1.59831i	−0.434584	−	0.900631i	\(-0.643105\pi\)
							0.716409	−	0.697680i	\(-0.245784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.27.4	✓	42
3.2	odd	2		981.2.w.a.136.4		42
109.105	even	9	inner	109.2.f.a.105.4	yes	42
327.323	odd	18		981.2.w.a.541.4		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.27.4	✓	42	1.1	even	1	trivial
109.2.f.a.105.4	yes	42	109.105	even	9	inner
981.2.w.a.136.4		42	3.2	odd	2
981.2.w.a.541.4		42	327.323	odd	18