Embedded newform 109.2.f.a.16.6

• Label: 109.2.f.a.16.6
• Level: $109$
• Weight: $2$
• Character: 109.16
• 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			16.6
Character	\(\chi\)	\(=\)	109.16
Dual form			109.2.f.a.75.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05944 + 1.83500i) q^{2} +(-1.78471 + 1.49755i) q^{3} +(-1.24481 + 2.15608i) q^{4} +(-0.222139 - 1.25981i) q^{5} +(-4.63879 - 1.68838i) q^{6} +(0.247600 + 1.40421i) q^{7} -1.03746 q^{8} +(0.421589 - 2.39095i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05944	+	1.83500i	0.749135	+	1.29754i	0.948238	+	0.317562i	\(0.102864\pi\)
							−0.199102	+	0.979979i	\(0.563803\pi\)
\(3\)	−1.78471	+	1.49755i	−1.03040	+	0.864610i	−0.990899	−	0.134610i	\(-0.957022\pi\)
							−0.0395032	+	0.999219i	\(0.512578\pi\)
\(5\)	−0.222139	−	1.25981i	−0.0993436	−	0.563405i	−0.993330	−	0.115310i	\(-0.963214\pi\)
							0.893986	−	0.448095i	\(-0.147897\pi\)
\(7\)	0.247600	+	1.40421i	0.0935839	+	0.530741i	0.995172	+	0.0981455i	\(0.0312910\pi\)
							−0.901588	+	0.432595i	\(0.857598\pi\)
\(11\)	2.41194	+	2.02386i	0.727226	+	0.610215i	0.929374	−	0.369140i	\(-0.120348\pi\)
							−0.202148	+	0.979355i	\(0.564792\pi\)
\(13\)	−0.714346	−	4.05126i	−0.198124	−	1.12362i	−0.907899	−	0.419188i	\(-0.862315\pi\)
							0.709776	−	0.704428i	\(-0.248796\pi\)
\(17\)	−0.508278	−	0.880364i	−0.123276	−	0.213520i	0.797782	−	0.602946i	\(-0.206007\pi\)
							−0.921058	+	0.389426i	\(0.872673\pi\)
\(19\)	1.95559	+	3.38717i	0.448642	+	0.777071i	0.998298	−	0.0583202i	\(-0.0185744\pi\)
							−0.549656	+	0.835391i	\(0.685241\pi\)
\(23\)	2.69078	−	4.66056i	0.561066	−	0.971794i	−0.436338	−	0.899783i	\(-0.643725\pi\)
							0.997404	−	0.0720115i	\(-0.0229418\pi\)
\(29\)	−8.13473	+	6.82585i	−1.51058	+	1.26753i	−0.647963	+	0.761672i	\(0.724379\pi\)
							−0.862618	+	0.505856i	\(0.831177\pi\)
\(31\)	1.50147	−	8.51524i	0.269671	−	1.52938i	−0.485723	−	0.874113i	\(-0.661444\pi\)
							0.755395	−	0.655270i	\(-0.227445\pi\)
\(37\)	1.19572	−	6.78125i	0.196575	−	1.11483i	−0.713584	−	0.700570i	\(-0.752929\pi\)
							0.910158	−	0.414260i	\(-0.135960\pi\)
\(41\)	−0.845707			−0.132077			−0.0660387	−	0.997817i	\(-0.521036\pi\)
							−0.0660387	+	0.997817i	\(0.521036\pi\)
\(43\)	−3.41014	−	5.90653i	−0.520041	−	0.900738i	−0.999729	−	0.0232985i	\(-0.992583\pi\)
							0.479687	−	0.877440i	\(-0.340750\pi\)
\(47\)	−8.66865	+	3.15513i	−1.26445	+	0.460223i	−0.885261	−	0.465095i	\(-0.846020\pi\)
							−0.379192	+	0.925318i	\(0.623798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.16.6	✓	42
3.2	odd	2		981.2.w.a.343.2		42
109.75	even	9	inner	109.2.f.a.75.6	yes	42
327.293	odd	18		981.2.w.a.838.2		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.16.6	✓	42	1.1	even	1	trivial
109.2.f.a.75.6	yes	42	109.75	even	9	inner
981.2.w.a.343.2		42	3.2	odd	2
981.2.w.a.838.2		42	327.293	odd	18